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FlyBy Math

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  Document Type: Lesson Plan
  Lesson Plan Type: Interactive Instruction
  Subject: Mathematics
  Grade Level: 5,6,7,8,9
  Time: 1-1.5 hours
  Last Updated: 05-26-2010
     
  Keywords:
     
     
 
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NASA
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CALIFORNIA STATE STANDARDS ADDRESSED

Mathematics/5/Algebra and Functions
1.1 Use information taken from a graph or equation to answer questions about a problem situation.
1.5 Solve problems involving linear functions with integer values; write the equation; and graph the resulting ordered pairs of integers on a grid.

Mathematics/5/Mathematical Reasoning
1.6 Analyze problems by identifying relationships, distinguishing relevant from irrelevant information, sequencing and prioritizing information, and observing patterns.
2.1 Use estimation to verify the reasonableness of calculated results.
2.2 Apply strategies and results from simpler problems to more complex problems.
2.3 Use a variety of methods, such as words, numbers, symbols, charts, graphs, tables, diagrams, and models, to explain mathematical reasoning.
2.4 Express the solution clearly and logically by using the appropriate mathematical notation and terms and clear language; support solutions with evidence in both verbal and symbolic work.
2.6 Make precise calculations and check the validity of the results from the context of the problem.
3.1 Evaluate the reasonableness of the solution in the context of the original situation.
3.2 Note the method of deriving the solution and demonstrate a conceptual understanding of the derivation by solving similar problems.
3.3 Develop generalizations of the results obtained and apply them in other circumstances.

Mathematics/5/Statistics Data Analysis and Probability
1.2 Organize and display single-variable data in appropriate graphs and representations (e.g., histogram, circle graphs) and explain which types of graphs are appropriate for various data sets.
1.5 Know how to write ordered pairs correctly; for example, ( x, y ).

Mathematics/6/Algebra and Functions
2.2 Demonstrate an understanding that rate is a measure of one quantity per unit value of another quantity.
2.3 Solve problems involving rates, average speed, distance, and time.

Mathematics/6/Mathematical Reasoning
1.1 Analyze problems by identifying relationships, distinguishing relevant from irrelevant information, identifying missing information, sequencing and prioritizing information, and observing patterns.
1.2 Formulate and justify mathematical conjectures based on a general description of the mathematical question or problem posed.
2.1 Use estimation to verify the reasonableness of calculated results.
2.2 Apply strategies and results from simpler problems to more complex problems.
2.3 Estimate unknown quantities graphically and solve for them by using logical reasoning and arithmetic and algebraic techniques.
2.4 Use a variety of methods, such as words, numbers, symbols, charts, graphs, tables, diagrams, and models, to explain mathematical reasoning.
2.5 Express the solution clearly and logically by using the appropriate mathematical notation and terms and clear language; support solutions with evidence in both verbal and symbolic work.
2.7 Make precise calculations and check the validity of the results from the context of the problem.
3.1 Evaluate the reasonableness of the solution in the context of the original situation.
3.2 Note the method of deriving the solution and demonstrate a conceptual understanding of the derivation by solving similar problems.
3.3 Develop generalizations of the results obtained and the strategies used and apply them in new problem situations.

Mathematics/6/Number Sense
1.2 Interpret and use ratios in different contexts (e.g., batting averages, miles per hour) to show the relative sizes of two quantities, using appropriate notations ( a/b, a to b, a:b ).

Mathematics/7/Algebra and Functions
1.5 Represent quantitative relationships graphically and interpret the meaning of a specific part of a graph in the situation represented by the graph.
3.3 Graph linear functions, noting that the vertical change (change in y- value) per unit of horizontal change (change in x- value) is always the same and know that the ratio ("rise over run") is called the slope of a graph.
3.4 Plot the values of quantities whose ratios are always the same (e.g., cost to the number of an item, feet to inches, circumference to diameter of a circle). Fit a line to the plot and understand that the slope of the line equals the quantities.
4.2 Solve multi step problems involving rate, average speed, distance, and time or a direct variation.

Mathematics/7/Mathematical Reasoning
1.4 Analyze problems by identifying relationships, distinguishing relevant from irrelevant information, identifying missing information, sequencing and prioritizing information, and observing patterns.
1.5 Formulate and justify mathematical conjectures based on a general description of the mathematical question or problem posed.
2.1 Use estimation to verify the reasonableness of calculated results.
2.2 Apply strategies and results from simpler problems to more complex problems.
2.3 Estimate unknown quantities graphically and solve for them by using logical reasoning and arithmetic and algebraic techniques.
2.5 Use a variety of methods, such as words, numbers, symbols, charts, graphs, tables, diagrams, and models, to explain mathematical reasoning.
2.6 Express the solution clearly and logically by using the appropriate mathematical notation and terms and clear language; support solutions with evidence in both verbal and symbolic work.
2.8 Make precise calculations and check the validity of the results from the context of the problem.
3.1 Evaluate the reasonableness of the solution in the context of the original situation.
3.2 Note the method of deriving the solution and demonstrate a conceptual understanding of the derivation by solving similar problems.
3.3 Develop generalizations of the results obtained and the strategies used and apply them to new problem situations.

Mathematics/7/Measurement and Geometry
1.2 Construct and read drawings and models made to scale.
1.3 Use measures expressed as rates (e.g., speed, density) and measures expressed as products (e.g., person-days) to solve problems; check the units of the solutions; and use dimensional analysis to check the reasonableness of the answer.

Mathematics/9/Algebra I
6.1 Students graph a linear equation and compute the x- and y- intercepts (e.g., graph 2x + 6y = 4). They are also able to sketch the region defined by linear inequality (e.g., they sketch the region defined by 2x + 6y < 4).
15.1 Students apply algebraic techniques to solve rate problems, work problems, and percent mixture problems.
 
BRIEF DESCRIPTION
This set of activities uses real-life applications of mathematics: students predict air traffic conflicts as air traffic controllers, pilots, engineers, and NASA scientists.  There are five different problems for students to solve, each building upon the previous one and using the same parameters: starting distances, speeds, and separation distances. 
 
PROCEDURES
 
Goal(s):
The FlyBy Math curriculum materials have two overarching goals:
  • To enable students to use mathematical reasoning and scientific inquiry to investigate and solve problems based on real-life scenarios.
  • To offer students a variety of problem-solving tools and approaches, ranging from experiments to paper-and-pencil activities.
 
Specific Objectives:
The five problems map to five different learning objectives.  Students will determine...
  1. If two planes are traveling at the same constant (fixed) speed on two different routes and the planes are the same distance from the point where the two routes come together, the planes will arrive at the intersection at the same time. So the planes will meet at the point where the routes come together.
  2. If two planes are traveling at the same constant (fixed) speed on two different routes and the planes are different distances from the point where the two routes come together, the planes will arrive at the intersection at different times. So the planes will not meet at the point where the routes come together.  Also, since the planes are traveling at the same constant (fixed) speed, the plane closest to the intersection will maintain its “headstart.” So at the intersection, the separation between the planes will be the same as the “headstart” of the plane that was closest to the intersection at the beginning of the problem.
  3. If two planes are traveling at different constant (fixed) speeds on two different routes and the planes are each the same distance from the point where the two routes come together, the planes will arrive at the intersection at different times. So the planes will not meet at the point where the routes come together.  Also, since the planes are traveling at different constant (fixed) speeds, their separation distance at the intersection is directly proportional to the difference in speeds. So, for example, if the difference in speeds were twice as great, then the separation at the intersection would also be twice as great.
  4. If two planes are traveling at different constant (fixed) speeds on two different routes and the planes are each a different distance from the point where the two routes come together, students must know those values in order to determine the separation distance between the planes at the intersection.
  5. If two planes are traveling at different speeds on the same route and the trailing plane is traveling faster than the leading plane, the trailing plane will close the gap at a rate equal to the difference in the speeds of the planes.  So if the difference in speeds is twice as great and the starting distance between the planes remains the same as the original starting separation distance, then the trailing plane will close the gap at twice the original rate. Therefore, the amount of time for the trailing plane to catch up to the leading plane will be half as great.  If the planes each travel at their original speeds but the starting distance between the planes is twice as great, then the trailing plane must now close twice the distance. However, the plane is traveling at the original rate. So the amount of time will double for the trailing plane to catch up to the leading plane.
 
Required Materials:
Student handouts:
  • Student Workbook
  • Assessment Package (optional)
Materials for the experiment:
  • sidewalk chalk or masking tape or cashier's tape or a knotted rope
  • measuring tape or ruler
  • marking pens (optional)
  • 1 stopwatch or 1 watch with a sweep second hand or 1 digital watch that indicates seconds
  • pencils
  • signs (available on the FlyBy Math website) identifying pilots, controllers, and NASA scientists
  • clipboard (optional)
 
Anticipatory Set (Lead-in):
To help your students understand the problem, you can ask them to consider this related problem that is set in a more familiar context.  For instance:
Two students, Ana and Alex, plan to meet at the movies. Each student lives the same distance from the theater. Ana and Alex will each leave their homes at the same time and walk at the same constant (fixed) speed.

You can ask your students these questions:

Will Ana and Alex arrive at the movie theater at the same time? Why or why not?

In particular, if your students think Ana and Alex will arrive at different times, ask them who will arrive first and why.

Since the students are walking at the same constant (fixed) speed and each must travel the same distance to the theater, students may realize that the students will arrive at the theater at the same time.
 
Lesson Plan Procedure:
1. What's FlyBy Math? Watch this introduction to air traffic control:



2. Watch this animation of 24 hours of flights in the US:



3. Watch this video of an air traffic controller on the job:



4. Select a problem to begin with. Most teachers begin with Problem 2.  Here are the links to each of the problems:
5. Administer the pretest (optional, see the Assessment Package) for the selected problem.

6. Give students the Student Workbook for the chosen problem and have the students READ THE PROBLEM and SET UP and DO THE EXPERIMENT.  In this small-group activity, students set up the experiment by marking off jet routes on the classroom floor or on an outdoor area.  To conduct the experiment, students assume the roles of pilots, air traffic controllers, and NASA scientists.  The pilots step down the jet route at a prescribed pace.  The air traffic controllers set the pace and record the time when the first plane arrives at the intersection (ATC Problems 1-4) or when the trailing plane overtakes the leading plane (ATC Problem 5).  The NASA scientists track and record the pilots' distances along the jet route.

7. Problem-solving methods may include counting feet and seconds, drawing and stacking blocks, plotting points on vertical lines, plotting points on a grid, deriving and using the distance-rate-time formula, or graphing two linear equations.
 
Closure (Reflect Anticipatory Set):
 Each workbook includes a section titled "Analyze Your Results."
 
Assessments & notes
 
Plan for Independent Practice:
 
 
Assessment Based on Objectives:
Administer the post test (optional, see the Assessment Package).
 
Possible Connections to Other Subjects:
 
 
Adaptations & Extensions:
Problems 2-5 have extensions that students can do (back of the Student Workbooks).
 
Additional Notes:
 None.
 
Copyright:
 
 
 
 
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Submitted by NASA on Fri, 04/02/2010 - 04:37


Title:

FlyBy Math

Grade Level:

5,6,7,8,9

Subject:

Mathematics

Author:

NASA

Time:

1-1.5 hours

Lesson Plan Type:

Interactive Instruction

Keywords:

distance, rate, time, drt, tables, graphs, linear equations, coordinate graphing, predicting outcomes, nasa, air traffic control, smart skies, nasa, fly by math, flyby math

Brief Description:

This set of activities uses real-life applications of mathematics: students predict air traffic conflicts as air traffic controllers, pilots, engineers, and NASA scientists.  There are five different problems for students to solve, each building upon the previous one and using the same parameters: starting distances, speeds, and separation distances. 

California State Standards Addressed:

Mathematics/5/Algebra and Functions)1.1,1.5
Mathematics/5/Mathematical Reasoning)1.6,2.1,2.2,2.3,2.4,2.6,3.1,3.2,3.3
Mathematics/5/Statistics Data Analysis and Probability)1.2,1.5
Mathematics/6/Algebra and Functions)2.2,2.3
Mathematics/6/Mathematical Reasoning)1.1,1.2,2.1,2.2,2.3,2.4,2.5,2.7,3.1,3.2,3.3
Mathematics/6/Number Sense)1.2
Mathematics/7/Algebra and Functions)1.5,3.3,3.4,4.2
Mathematics/7/Mathematical Reasoning)1.4,1.5,2.1,2.2,2.3,2.5,2.6,2.8,3.1,3.2,3.3
Mathematics/7/Measurement and Geometry)1.2,1.3
Mathematics/9/Algebra I)6.1,15.1

Related Links:

Link 1:

Goal(s):

The FlyBy Math curriculum materials have two overarching goals:
  • To enable students to use mathematical reasoning and scientific inquiry to investigate and solve problems based on real-life scenarios.
  • To offer students a variety of problem-solving tools and approaches, ranging from experiments to paper-and-pencil activities.

Specific Objectives:

The five problems map to five different learning objectives.  Students will determine...
  1. If two planes are traveling at the same constant (fixed) speed on two different routes and the planes are the same distance from the point where the two routes come together, the planes will arrive at the intersection at the same time. So the planes will meet at the point where the routes come together.
  2. If two planes are traveling at the same constant (fixed) speed on two different routes and the planes are different distances from the point where the two routes come together, the planes will arrive at the intersection at different times. So the planes will not meet at the point where the routes come together.  Also, since the planes are traveling at the same constant (fixed) speed, the plane closest to the intersection will maintain its “headstart.” So at the intersection, the separation between the planes will be the same as the “headstart” of the plane that was closest to the intersection at the beginning of the problem.
  3. If two planes are traveling at different constant (fixed) speeds on two different routes and the planes are each the same distance from the point where the two routes come together, the planes will arrive at the intersection at different times. So the planes will not meet at the point where the routes come together.  Also, since the planes are traveling at different constant (fixed) speeds, their separation distance at the intersection is directly proportional to the difference in speeds. So, for example, if the difference in speeds were twice as great, then the separation at the intersection would also be twice as great.
  4. If two planes are traveling at different constant (fixed) speeds on two different routes and the planes are each a different distance from the point where the two routes come together, students must know those values in order to determine the separation distance between the planes at the intersection.
  5. If two planes are traveling at different speeds on the same route and the trailing plane is traveling faster than the leading plane, the trailing plane will close the gap at a rate equal to the difference in the speeds of the planes.  So if the difference in speeds is twice as great and the starting distance between the planes remains the same as the original starting separation distance, then the trailing plane will close the gap at twice the original rate. Therefore, the amount of time for the trailing plane to catch up to the leading plane will be half as great.  If the planes each travel at their original speeds but the starting distance between the planes is twice as great, then the trailing plane must now close twice the distance. However, the plane is traveling at the original rate. So the amount of time will double for the trailing plane to catch up to the leading plane.

Required Materials:

Student handouts:
  • Student Workbook
  • Assessment Package (optional)
Materials for the experiment:
  • sidewalk chalk or masking tape or cashier's tape or a knotted rope
  • measuring tape or ruler
  • marking pens (optional)
  • 1 stopwatch or 1 watch with a sweep second hand or 1 digital watch that indicates seconds
  • pencils
  • signs (available on the FlyBy Math website) identifying pilots, controllers, and NASA scientists
  • clipboard (optional)

Anticipatory Set (Lead-in):

To help your students understand the problem, you can ask them to consider this related problem that is set in a more familiar context.  For instance:
Two students, Ana and Alex, plan to meet at the movies. Each student lives the same distance from the theater. Ana and Alex will each leave their homes at the same time and walk at the same constant (fixed) speed.

You can ask your students these questions:

Will Ana and Alex arrive at the movie theater at the same time? Why or why not?

In particular, if your students think Ana and Alex will arrive at different times, ask them who will arrive first and why.

Since the students are walking at the same constant (fixed) speed and each must travel the same distance to the theater, students may realize that the students will arrive at the theater at the same time.

Lesson Plan Procedure:

1. What's FlyBy Math? Watch this introduction to air traffic control:



2. Watch this animation of 24 hours of flights in the US:



3. Watch this video of an air traffic controller on the job:



4. Select a problem to begin with. Most teachers begin with Problem 2.  Here are the links to each of the problems:
5. Administer the pretest (optional, see the Assessment Package) for the selected problem.

6. Give students the Student Workbook for the chosen problem and have the students READ THE PROBLEM and SET UP and DO THE EXPERIMENT.  In this small-group activity, students set up the experiment by marking off jet routes on the classroom floor or on an outdoor area.  To conduct the experiment, students assume the roles of pilots, air traffic controllers, and NASA scientists.  The pilots step down the jet route at a prescribed pace.  The air traffic controllers set the pace and record the time when the first plane arrives at the intersection (ATC Problems 1-4) or when the trailing plane overtakes the leading plane (ATC Problem 5).  The NASA scientists track and record the pilots' distances along the jet route.

7. Problem-solving methods may include counting feet and seconds, drawing and stacking blocks, plotting points on vertical lines, plotting points on a grid, deriving and using the distance-rate-time formula, or graphing two linear equations.

Closure (Reflect Anticipatory Set):

 Each workbook includes a section titled "Analyze Your Results."

Plan for Independent Practice:

 

Assessment Based on Objectives:

Administer the post test (optional, see the Assessment Package).

Possible Connections to Other Subjects:

 

Adaptations and Extensions:

Problems 2-5 have extensions that students can do (back of the Student Workbooks).

Additional Notes:

 None.

»