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A Smart Skies TM Product

Math-Based Decisions in Air Traffic Control

Grades 5–9

Interactive Simulator Workbook

An Educator Guide to

Educational Product

Educators

& Students Grades 5-9
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Skies TM

SmartSkiesTM LineUp With MathTM was developed and produced at NASA Ames 

Research Center with sponsorship by the Airspace Systems Program and with 

contributions and support from the following individuals:

Robert Jacobsen, Program Manager, NASA Airspace Systems Program

Gregory W. Condon, Project Lead, Smart Skies Education Project

Miriam Landesman, Education Lead, Consultant–Math, Education & Technology

Stacey Carpenter, Teacher, Oakland Unified School District

William Preston, Nancy Smith, Lynne Martin, Tom Prevot, Bruce Heath, Crystal Harris, ShaRhonda Kersey

Lydia Mackovitch, Khanh Trinh

William Preston, Forrest West

Rebecca Sims, Christina O’Guinn

Mark Cavolowsky, Christopher Cho

Linda Conrad, Jeffrey Simmons, Geoffrey Bruce, Christopher Cho

Jesse Carpenter and the NASA Ames Video Group, Lee Hall and the FAA Oakland 

ARTCC staff, Jake Burkart

Dr. Dallas Denery

The authors thank the teachers and students who tested LineUp With MathTM including the following teachers who participated in the local classroom tests:

JoAnne Asato, Larry Cohbra, Martha Hanisch, Agnes Kaiser, Arthur Michelson, 

Elizabeth Mogin, Dolores Procaccio, Bernadette Salgarino

All materials are free and available to download from the LineUp With MathTM websites:

Teacher website: https://www.nasa.gov/lineup-with-math

Student (Simulator): https://atcsim.nasa.gov

LineUp With MathTM is also available in electronic format through the NASA Education Portal—

NASA’s electronic resources specifically developed for the educational community.

https://www.nasa.gov/audience/foreducators/index.html

This publication is in the public domain and is not protected 

by copyright. Permission is not required for duplication.

Sponsor

Principal Authors

Teachers and Students

Technical Review

Movie Support

Web and Graphic Support

Strategies and Solutions

Education Evaluation

ATC Technical Support

Software Development

Simulator Concept

ACKNOWLEDGEMENTS
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BECOMING ACQUAINTED WITH LineUp With MathTM 1

What is LineUp With MathTM ? 1

Access to Materials 

Quick Start Guide 

Background  

Standards  

Overview 

6 Problem Sets 

ATC Simulator 

Background and Instructional Movies 

Student Workbooks .

Teachers Guides 

Instructional Goals 

PREPARING TO TEACH LineUp With MathTM 1

Select Materials  

Review Materials 

IMPLEMENTING LineUp With MathTM WITH YOUR STUDENTS 7

Instructional Activities 

Materials 

NASA RESOURCES FOR EDUCATORS 8

PROVIDING NASA WITH YOUR ASSESSMENT .

APPENDICES 9

Appendix I - LineUp With MathTM Website Home Page 

Appendix II - Glossary .

TABLE OF CONTENTS
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Educator Guide for LINEUP WITH MATHTM

LineUp With MathTM enables students to explore and apply decision-making and 

proportional reasoning skills to resolve distance-rate-time problems in realistic air traffic 

control situations. To solve the problems, students use a web-based interactive Air Traffic 

Control (ATC) Simulator that represents an air traffic controller’s screen. The accompanying Workbooks provide the underlying mathematics and strategies to enable students 

to optimize their solutions.

LineUp With MathTM consists of six Problem Sets. Each typically contains:

• interactive ATC Simulator problems (web-based)

• A Student Workbook (print-based)

• supporting teacher materials

All LineUp With MathTM materials, including the interactive ATC Simulator, are free 

and accessible from the LineUp With MathTM teacher website: (See Appendix I for a brief 

introduction to the website.)

https://www.nasa.gov/lineup-with-math

Students access to the ATC Simulator is available at the following address:

http://atcsim.nasa.gov

For a brief introduction to the LineUp With MathTM materials and ways to use these materials in your classroom, see the LineUp With MathTM Quick Start Guide available on the 

teacher website.

LineUp With MathTM was developed under Smart SkiesTM, a key part of the NASA 

Airspace Systems Program’s educational efforts. The Program’s goal is to develop advanced 

computer-based systems to help pilots and air traffic controllers operate the nation’s air 

transportation system with reduced flight delays and improved efficiency and access.

In the first Smart SkiesTM product, FlyBy MathTM, students conduct experiments and use 

a variety of math methods to analyze traditional distance-rate-time problems in air traffic 

control. As students identify conflicts in these 2-plane problems, they develop and apply 

proportional reasoning to predict distance-rate-time relationships.

State Standards Alignment I: For each of the 50 states, the alignment of LineUp With Math- TM with the state mathematics standards is available on the LineUp With MathTM website.

National Standards Alignment: The alignments of LineUp With MathTM with the National 

Council of Teachers of Mathematics (NCTM) Standards and Expectations and the 

National Science Education Standards (NSES) are available on the LineUp With MathTM

website.

BECOMING ACQUAINTED WITH

LINEUP WITH MATH 

TM

A Smart SkiesTM Educational Product

What is LineUp With 

MathTM?

Access to Materials

Quick Start Guide

Background

Standards

Smart

Skies TM
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In LineUp With MathTM, the second Smart SkiesTM product, students apply proportional 

reasoning to make decisions and resolve conflicts in realistic air traffic control problems 

involving two or more planes. The challenge in each problem is to “line up” the planes safely, 

with the proper spacing, at a given intersection of jet routes.

Students typically explore the problems using an interactive Air Traffic Control (ATC) 

Simulator. To increase and refine their decision-making skills, students use the print-based 

Student Workbooks that provide instruction and practice in mathematics and strategies to 

enable them to optimize their solutions. After completing the Student Workbooks, students 

return to the Simulator and apply their newly learned skills to properly line up the planes 

while also optimizing the time for the last plane to reach the final intersection.

LineUp With MathTM addresses the following types of ATC problems:

• two-plane and three-plane conflicts resolved with route changes

• two-plane and three-plane conflicts resolved with speed changes

• three-plane, four-plane, and five-plane conflicts resolved with speed changes or with 

speed and route changes.

LineUp With MathTM uses an airspace based upon a real sector in the National Airspace 

System. The sector, formally designated as ZOA33 and referred to as Sector 33, is located in 

the northern California. In LineUp With MathTM, the sector distances have been modified to 

simplify student calculations.

The LineUp With MathTM instructional materials have been packaged in the form of six 

Problem Sets.

• Problem Set A: An introduction to the environment of real air traffic control.

• Problem Sets B and C: Analysis of conflicts between two or three planes and 

resolution of the conflicts by rerouting.

• Problem Set D: An introduction to the basic effects of differences in speed.

• Problem Sets E and F: Analysis of conflicts between two or three planes and 

resolution of the conflicts by changing plane speed.

Each Problem Set contains:

• an interactive or multimedia component (e.g., Simulator, movies)

• a Student Workbook

• a Teacher Guide with answers to the Workbook and to the featured Simulator problems

Each of these components is described below.

Overview

6 Problem Sets

Before After
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The interactive ATC Simulator is a web-based computer program that teachers can access 

from the LineUp With MathTM website or directly from a separate simulator website that 

gives students easy access to the ATC Simulator only (and not to the answers and solutions 

provided on the teacher website):

https://www.nasa.gov/lineup-with-math

The Simulator provides a user-friendly realistic simulation of Sector 33 as viewed by a 

controller on a sector display screen. At the beginning of each Simulator problem, two 

or more planes travel along merging jet routes. Students, preferably working in pairs or

individually, use proportional reasoning and distance-rate-time relationships to identify 

potential conflicts at the intersections of the jet routes. Depending upon the problem, 

students can resolve the conflict by changing routes or plane speeds.

As mentioned previously, the goal is to “line up” the planes safely, with proper spacing, at 

a given intersection. Students can rerun a problem to refine their solution and bring each 

plane to the intersection, not only safely, but also as quickly as possible. (The target time 

shown on the simulator screen for each problem is the optimal time for that problem.)

The Simulator features problems with 2, 3, 4, and 5 planes. The key two- and three-plane 

problems are supported by the Student Workbooks. The remaining two- and three-plane 

problems provide additional practice. The four- and five-plane problems enable students to 

explore more challenging scenarios.

Each Simulator problem is named with two numbers separated by a hyphen, e.g., 2-5. The 

first number, 2, corresponds to the number of planes in the problem. The second number, 

5, indicats the problem is the fifth in the sequence of 2-plane problems.

The simulator website also includes the following user support materials:

• Animated tutorial

• 2-page quick reference guide

• technical requirements

LineUp With MathTM includes three movies available for download on the LineUp With 

MathTM teacher website and on the student Simulator website.

The first video, “Animation of 24 hours of flight in the U.S.”, compresses 24-hours of 

US flight paths to 1 minute. The video illustrates the world’s biggest distance-rate-time 

problem and motivates the study of air traffic control.

The second movie, “What is LineUp With MathTM?”, introduces students to the overall 

goals and activities, and features scenes of students engaged in LineUp With MathTM.

The third movie, “Welcome to Sector 33”, presents the vocabularly, units, and graphical 

representations used in air traffic control. It prepares students for their first session with 

the interactive ATC Simulator. It also prepares students for the activities in the first 

Student Workbook (Problem Set A).

ATC Simulator

Background and 

Instructional 

Movies
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For each Problem Set, a print-based Student Workbook provides a structured

learning environment. Each Workbook contains paper-and-pencil worksheets that

introduce students to pertinent air traffic control concepts as well as problem analysis and 

solution methods.

• Problem Set A: Workbook A introduces students to the environment of real air 

traffic control.

• Problem Sets B and C: Students use Workbooks B and C to analyze conflicts

between two or three planes and to resolve the conflicts by rerouting.

• Problem Set D: Workbook addresses the basic effects of differences in speed.

• Problem Sets E and F: Students use Workbooks E and F to analyze conflicts between 

two or three planes and to resolve the conflicts by changing plane speed.

Workbooks B, C, E, and F include worksheets that support selected Simulator problems.

For these worksheets, the title matches the number problems enable students to explore 

more challenging scenarios.

Each of the six Problem Sets is accompanied by a Teacher Guide that includes:

• an overview

• a list of objectives, prerequisites, and materials

• a discussion and speciication of the relevant ATC Simulator Problems 

(or other interactive/multimedia components.

• a discussion of key points for each Simulator problem and for each Workbook worksheet

• the solutions for each Simulator problem

• the answers each Workbook worksheet.

LineUp With MathTM has two overarching goals:

• To enable students to use mathematical reasoning and problem-solving skills to

investigate and sole realistic distance-rate-time problems in air traffic control.

• To offer students a variety of problem solving approaches, tools, skills, and

experiences in the context of challenging, yet accessible, real-world problems.

The following table lists the materials and specific learning objectives for each LineUp 

With MathTM Problem Set.

Student Workbooks

Instructional Goals

Teacher Guides
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Problem Set Materials Objectives

A

Introduction to 

air traffic control 

sectors

• Three Movies

--Animation of 24 hours 

of flight in the U.S.

-- What is LineUp With 

Math?

--Welcome to Sector 33

• Student Workbook A

• Teacher Guide A

• Learn the vocabulary of air traffic control.

• Learn the units (nautical miles and knots) of air traffic control.

• Learn to read and interpret an airspace sector diagram.

• Learn the air traffic control spacing requirements for ssafety and 

efficiency.

B

2-plane conflicts: 

change routes

• Animated Tutorial -- 

Introduction to the ATC 

Simulator

• Simulator Problems 2-1*, 

2-2*, 2-3*, 2-9, 2-10

• Student Workbook B

• Teacher Guide B

• Analyze a sector diagram to identify a spacing conflict between 

two planes, each traveling at the same speed.

• Resolve the conflict by changing the route for one plane.

C

3-plane conflicts: 

change routes

• Simulator Problems 

3-1*, 3-2*, 3-7

• Student Workbook C

• Teacher Guide C

• Analyze a sector diagram to identify a spacing conflict among three 

planes, each traveling at the same speed.

• Resolve the conflict by changing the route for one or more planes.

D

Effects of speed 

differences

• Student Workbook D

• Teacher Guide D

• Learn that when a plane’s speed is reduced, the difference in the 

plane’s distance traveled can be calculated by multipying the 

difference in speed by the amount of time traveled.

• Learn how to associate a decrease in plane speed in knots with a 

decrease in distance traveled each minute.

• Learn the correspondence between a 60-knot reduction in plane 

speed and the resulting decrease in distance traveled (in 1 minute) 

over a 10 nautical mile interval on a jet route. (Students will 

encounter these particular speed reductions and multiples of 10 

nautical miles in subsequent Workbooks.)

E

2-plane conflicts: 

change speeds

• Simulator Problems

2-4*, 2-5*, 2-6*, 2-7*, 

2-8*, 2-11*, 2-12, 2-13, 

2-14, 2-15, 2-16

• Student Workbook E

• Teacher Guide E

• Analyze a sector diagram to identify spacing conflicts between two 

planes, each traveling at the same speed.

• Resolve the conflicts by changing the speed of one plane.

• (Optional) Learn that a given percent reduction in plane speed 

yields the same percent reduction in distance traveled in the original 

amount of time.

F

2-plane conflicts: 

change speeds

• Simulator Problems 

3-3*, 3-4*, 3-5*, 3-6*, 

3-8, 3-9, 3-10, 3-11, 

3-12

• Student Workbook F

• Teacher Guide F

• Analyze a sector diagram to identify spacing conflicts among three 

planes, each traveling at the same speed.

• Resolve conflicts by changing the speed of one or more planes.

• Resolve conflicts by changing the speed or the speed and route of 

one or more planes.

* The Simulator problems marked with an asterisk (*) are supported by worksheets in the print-based Student Workbooks.
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To prepare to teach LineUp With MathTM

1. Go to the LineUp With MathTM teacher website:

https://www.nasa.gov/lineup-with-math

2. On the website, view the movie “Animation of 24 hours of flight in the U.S.” to 

see the world’s largest distance-rate-time problem.

3. On the website, view the movie “What is LineUp With MathTM?”.

4. On the website, view the movie “Welcome to Sector 33” to see an overview of the 

vocabularly, units, and graphical representations of air traffic control problems. 

This material is the backbone of both the simulator and the print-based workbooks.

5. On the website, view the movie “Introduction to the ATC Simulator.”

6. Access and try the on-line Simulator. Online instructions are provided to assure 

that your computer is configured to support the Simulator.

7. Select the Problem Set you will use to introduce your students to LineUp With 

MathTM. We strongly recommend starting with Problem Set A.

8. Follow the online instructions to select and download the materials for the 

Problem Set you have selected.

9. With the student and teacher materials in hand, and with access to the ATC 

Simulator, read the following Implementation section in this document.

10. Duplicate the Student Workbook to provide each student with one copy.

11. Now you’re cleared for takeoff! 

Select Materials

Review Materials

PREPARING TO TEACH LINEUP WITH MATHTM

If you do not want to 

print the documents, 

you can simply preview 

them online.


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The following sequence of instructional activities is recommended for implementing each

LineUp With MathTM Problem Set. As you read this section, you may find it helpful to

have a copy of a Student Workbook and the associated teacher materials.

1. If your students are new to LineUp With MathTM, begin with the three introductory 

movies:

-- “An animation of 24 hours of flight in the U.S.”

-- “What is LineUp with MathTM?”

-- “Welcome to Sector 33.”

The movie clips are available on the LineUp With MathTM teacher website.

2. If your students are new to LineUp With MathTM, introduce the ATC Simulator. You 

may want students to view the animated “ATC Simulator User Tutorial.” You may 

choose to project the Simulator (and animated tutorial) for a whole class activity. 

Or you may choose to have the students work on individual computers, in pairs or

individually.

Students should access the Simulator and movies at the following address:

http://atcsim.nasa.gov

3. For the Problem Set you have selected, assign the Simulator problems featured in the 

Student Workbook. Provide time for your students to explore the Simulator and the 

assigned interactive ATC problems.

4. For the Problem Set you have selected, assign the worksheets in the accompanying 

Student Workbook.

5. After students have completed the Workbook, have them revisit the assigned problems 

on the Simulator to see if they can optimize their solution and match the target time 

on the Simulator screen. You may also wish to assign some of the supplementary Simulator problems. (See the table in the Instructional Goals section of this document or 

consult the Teacher Guide for the Problem Set.)

If you plan to teach a subsequent Problem Set, you may want to have your students 

experiment with the first few Simulator problems from that Set.

Classroom Materials:

-- Movie clips

-- ATC Simulator

-- Student Workbook for selected Problem Set

-- Simulator Quick Start Guide

Instructional Activities

Materials

IMPLEMENTING LINEUP WITH MATHTM

WITH YOUR STUDENTS

Estimated time: 

30 minutes

Estimated time: 

30 minutes

Estimated time: 

30 minutes

Estimated time: 

30 minutes

Estimated time: 

30 minutes
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Classroom Materials

-- Teacher Quick Start Guide

-- LineUp With MathTM Educator Guide (this document)

-- Teacher Guide for selected Problem Set with answers to all worksheets and 

solutions to featured Simulator problems

-- Complete set of solutions to all ATC Simulator problems

NASA RESOURCES FOR EDUCATORS

NASA has a rich supply of resources for educators and delivery methods ranging from 

print-based, multimedia, and television.

These resources can all be accessed from the NASA web portal at:

https://www.nasa.gov

Select the “for EDUCATORS” button to access the educational materials.

They may also be accessed from the Smart SkiesTM website at:

http://www.smartskies.nasa.gov

Select the Resources button.

PROVIDING NASA WITH YOUR ASSESSMENT

In order to ensure that these educational materials are the best they can be for teachers and 

students across the nation, NASA needs your feedback and suggestions.

Please take a few minutes to fill out the Educator Reply Card located under the CONTACT 

US button on the Smart SkiesTM website:

http://www.smartskies.nasa.gov

Thank you!
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APPENDICES

Appendix I - LINEUP WITH MATHTM Website Home Page

All Student

Materials

All Teacher

Materials

Product

Overview

Everything

you need for

a Problem Set

NCTM and

State Standards

Alignments

Access to Simulator

and Student Website
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Device(s) that are used or intended to be used for flight in the air, and when used 

in the air traffic control terminology, may include the flight crew.

Aircraft operating in the air or on an airport surface, exclusive of loading ramps 

and parking areas.

The space lying above the earth or above a certain area of land or water, especially 

the space lying above a nation and coming under its jurisdiction.

A service operated by appropriate authority to promote the safe, orderly and 

expeditious flow of air traffic.

The unique designator for each flight. Is composed of a 3-letter abbreviation for 

the airline name and a unique flight number.

When 2 aircraft do not meet the minimum separation standards.

Specified information relating to the intended flight of an aircraft that is filed 

orally or in writing with an FSS or an ATC facility.

A line, course, or track along which an aircraft is flying or intended to be flown.

In air traffic control, the distance between aircraft to assure their orderly 

movement.

Units of speed for air and sea travel. One knot equals one nautical mile per hour.

In air traffic control, the minimum allowed distance between aircraft to achieve 

their safe movement in flight and while landing and taking off.

Unit of measure for distance in air and sea travel. A nautical mile is 1.15 times a 

statute mile used in land travel.

A defined path, consisting of one or more courses in a horizontal plane, which 

aircraft traverse over the surface of the earth.

A geographic area of the airspace for which a controller is responsible.

Aircraft

Air Traffic

Sector

Route

Nautical Miles

Minimum Separation

Knot

Ideal Spacing

Flight Path

Flight Plan

Conflict

Call Sign

Air Traffic Control

Airspace

Appendix II – Glossary
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National Aeronautics and Space Administration

Ames Research Center

Moffet Field, CA 94035