Mathematics Senior Level Capstone Course
Unit Overview
Title of Unit: Crash Landing Unit Designers:
(Name and School Division) Stephanie Brady, Hanover County Lauren Noehl, York County Diane Leighty, Editor, UVA-SCPS Office of Mathematics Outreach Context:
Summary of the issue, challenge, investigation, or problem.
An airplane is experiencing a mechanical failure and is descending at a constant rate of speed. Your task is to determine how far the plane will fly before it must land, and to find a safe place for the plane to come down. Number of Class
Hours: 3- 4 hours
Unit
Design: __x_Task Based ___Project Based
Other Subject Areas/Disciplines Addressed:
Travel
Journalism/Reporting Geography/Topology
Driving Question: How can an air traffic controller use information about a flight to make predictions about the path a plane is on? How can this information help the controller find safe places for a plane to land?
Mathematics Content Addressed:
Multiple representations of functions for analysis and prediction. Investigate relationships between zeros of a function and x-intercepts. Investigate and analyze linear functions. Graph linear equations, determine the slope of a line and write the equation of a line. Use similar geometric objects to solve real world problems.
MPE Addressed:
Understanding and Applying Functions
Assumption of Prior Knowledge:
Knowledge of linear functions, slope, equations of lines in point-slope form, x- and y-intercepts, zeros of a function.
College and Career Readiness/21st Century Skills to be taught (T) during this unit or expectation (E) for
Collaboration –Students will work in pairs. E & A Research –Find maps to determine landing areas.
E & A Communication (Written)
Written report for formal summative
E & A Technology – Use of graphing calculator to analyze and create model, use of Google
student use during this unit and assessed (A): BIE Page 35-37
assessment, short “press release” about landing to share with peers.
Maps or other online mapping tool, use of word processor to produce report.
Critical Thinking/Decision Making – Analysis of mathematical model to make a decision.
T & A Other: (Describe)
Major Products and/or
Performances: Group - Written report showing appropriate mathematical model, choice of landing location from a selection of options. Presentation Audience:x Class School
Individual - Mathematician’s Journal entries x Expert – News personnel option Community
Other: Launch: Event or
experience used to engage the students interest and inquiry:
Students are presented with the story about the plane. Video clip from of plane crashes in different areas. http://www.youtube.com/watch?v=jZPvVwvX_Nc (Hudson river simulation)
http://www.youtube.com/watch?v=Z7TDXd4t_v4 (belly landing in Poland) http://www.youtube.com/watch?v=CUkZevSbZ38 (another belly landing)
http://www.youtube.com/watch?v=UcI-pMC1ALA&feature=related (Delta flight has cracked windshield; emergency landing at Midway)
Evaluation: Formative Assessments (During the Unit)
Interview x Practice Presentations
Mathematicians Journal x Notes x
Preliminary
Plans/Outlines/Prototypes
x Checklists
Rough Drafts x Concept maps
Field Tests Other: Parts c, d and e
of Explore task
x Summative Assessment
(End of Project)
Written Products, with a rubric x Peer Evaluation, with a rubric
Other Product(s) or
Performance(s), with a rubric
Other: Press Release feedback in journal entries
x
Resources Needed: On-site people, facilities: Teacher
Equipment/Technology: Computer with internet access and word processing, graphing calculator Materials: Poster board or other presentation material for press release.
Community Resources: None.
Reflection Methods: Individual, Group, and/or Whole Class
Mathematicians Journal x Small/Focus Groups Whole Class Discussions x Fishbowl Discussions
Survey Other:
Material Adapted From:
Explore tasks and airplane problem based on tasks from http://www.exter.edu/academics/72_6539.aspx
Virginia’s Senior Level Capstone Course Instructional Plan
Unit Title: Crash Landing
Task: An airplane is flying above Nebraska when both engines fail and the plane starts going down. You are a traffic controller and need to find them a safe place to land.
ENGAGE How will student’s interested be peaked so they want to engage in the inquiry in this unit? Number of hours 20 min
Video clip from of plane crashes in different areas. Choose one or more to show.
http://www.youtube.com/watch?v=jZPvVwvX_Nc (Hudson river simulation)
http://www.youtube.com/watch?v=Z7TDXd4t_v4 (belly landing in Poland)
http://www.youtube.com/watch?v=CUkZevSbZ38 (another belly landing)
http://www.youtube.com/watch?v=UcI-pMC1ALA&feature=related (Delta flight has cracked windshield; emergency landing at Midway)
Whole class discussion of video clips – if your plane was going down, where would you want to land? What would you need to know about the area, plane, weather, etc?
Mathematician Journal Prompts Where would you like your plane to crash? Explain your reasoning.
EXPLORE Teacher provides guidance for the explorations to prepare students with the
knowledge and skills to engage in the task. Students will self-assess on the prior knowledge and skills assumed for the unit
Have students analyze slope in a point-slope form. See how different slopes affect x- and y-intercepts.
Positive Connections:
The equation y – 5 = m(x – 2) represents a line, no matter what value m has.
(a) What do all these lines have in common? **Whole class discussion after journal – What about the equation and line never change? What does change?
How does that change affect the line?
(b) When m = -2, what are the x- and y-intercepts?
**Teacher leads the discussion with student input. Discuss where intercepts are located, what
positive/negative/zero/undefined slope look like (c) When m = -1/3, what are the x- and y-intercepts?
Mathematician Journal Prompts What about the equation and line never change?
What does change?
Number of hours:
Journal: 10 min Discussion: 15 min
(b-e): 25 min
(d) When m = 2, what are the x- and y-intercepts?
(e) For what values of m are the axis intercepts both positive? **Students explore with a partner, turn in work and written explanation for formative assessment
**Teacher may suggest: Draw a picture of a line that fits and one that doesn’t, what conclusions can you draw from this? At what point does right turn into wrong? Keep in mind that the origin does not count as both positive. Students must be able to manipulate an equation in point-slope form to find intercepts. They understand how positive and negative slopes affect intercepts. Students are encouraged to use multiple representations.
Materials: graph paper Rubrics: attached – HO # 5
Self-Assessment/Tutorials attached – HO # 1-2 Teacher Solutions – HO #6
EXPLAIN Teacher introduces the main task of the unit and prepares students to in small groups.
Number of Hours: 20 min
Introduce the problem:
An airplane is flying at 36,000 feet directly above Lincoln, Nebraska. Both engines fail and the plane starts going down. A little later the plane is at 28,000 feet directly above Des Moines, Iowa, which is 160 miles from Lincoln. Assuming a constant rate of decent, predict how far from Des Moines the airplane will be when it crashes.
You are an air traffic controller and need to find a safe place for the plane to land. Based on the geographic location, does the plane need to change course to crash in a more ideal spot? How would it have to change? Where will you land it?
Students work with a partner, a group of 3 if necessary. Student task sheet is attached – HO # 3
Student task sheet modified for diversity – HO # 4
Mathematician Journal Prompts What research will you need to do to land the plane? ELABORATE The student groups are working independently with teacher consultations.
Working in pairs, students calculate distance left before plane hits the ground (use point-slope and graph). Students research the possible locations at that distance from the plane’s current position and decide on the “best” location to land the plane; then write an explanation for choosing that location.
Students create a “press release” for the plane landing. They can choose their own form of media representation (i.e:
Number of Hours: 1 hour
newspaper/webpage/video clip/etc.). Student checklist/rubric: attached – HO # 5
Formative Assessments: Written explanation, Journal prompts
Are there other factors that could change the landing
location? EVALUATE
Working groups submit products or make
presentations Number of Hours 1 hour
2 minute presentations of “press release”.
Review others’ products on pros and cons. What would you choose to read/watch to gain information about this crash?
Presentation given to whole class. May invite news media personnel to watch, offer suggestions, and answer questions.
Mathematician Journal Prompts Do you think your press release is representative of the best form of media for this event?
Map the Unit
What do students need to know and be able to do to complete the task/project/problem
successfully? How and when will they assess their own necessary knowledge and skills? How will they remediate their own gaps or weaknesses in knowledge and skills? Look at each major task for the unit and analyze the tasks necessary to produce a high-quality product.
Task:
KNOWLEDGE AND SKILLS NEEDED Assumed
already learned
Students will self-assess
Will be taught during the unit 1. Multiple representations of functions X X
2. Analyze and predict X X
3. Relationship between zeros and intercepts
X X
4. Investigate and analyze linear function X
5. Graph linear equations X X
6. Determine slope of a line X X
7. Write the equation of a line in point-slope form.
X
What project tools will student’s use? Know/need to know lists Daily goal sheet
Mathematician’s Journals Briefs/Memos
Task lists
Planning Calendar
HO # 1
Student Recommendation for Tutorial and Practices 1. Knowledge of linear functions
http://www.khanacademy.org/math/algebra/algebra-functions/v/basic-linear-function http://www.khanacademy.org/math/algebra/algebra-functions/v/linear-function-graphs http://www.zweigmedia.com/ThirdEdSite/tutorialsf0/frames1_3.html
http://www.youtube.com/watch?v=qfI3DMiu7fE 2. Slope
http://www.khanacademy.org/math/algebra/linear-equations-and-inequalitie/v/slope-of-a-line http://www.purplemath.com/modules/slope.htm
http://www.mathwarehouse.com/algebra/linear_equation/slope-of-a-line.php
3. Equations of lines in point-slope form
http://easycalculation.com/analytical/learn-point-slope.php
http://www.khanacademy.org/math/algebra/linear-equations-and-inequalitie/v/point-slope-and-standard-form
http://www.videosurf.com/video/point-slope-form-of-the-equation-of-a-line-137472060 4. x- and y-intercepts
http://www.khanacademy.org/math/algebra/linear-equations-and-inequalitie/v/x--and-y-intercepts
http://www.analyzemath.com/Graphing/GraphingLinearFunction.html 5. Zeros of a function
http://www.khanacademy.org/math/algebra/quadtratics/v/ca-algebra-i--quadratic-roots http://www.analyzemath.com/function/zeros.html
http://www.cliffsnotes.com/study_guide/Quiz-Zeros-of-a-Function.topicArticleId-257309,articleId-257203.html
6. Equations of circles
http://www.regentsprep.org/Regents/math/algtrig/ATC1/circlelesson.htm http://www.mathsisfun.com/algebra/circle-equations.html
HO # 2 Name_______________________ Date__________ Crash Landing: Student self-assessment
Slope:
1. Imagine a plane taking off for a flight. Would you say that the slope of the line describing the plane’s height compared to time in the air is positive, negative, zero or undefined?
2. Imagine a helicopter taking off from the deck of a ship at sea, straight up into the air. Would you say that the slope of the line describing the helicopter’s height compared to time in the air is positive, negative, zero or undefined?
3. Given two points of data from a linear relation, (-100, 28,050) and (200, 50,000), what is the slope of the line?
Graphing Linear Functions:
4. Sketch a coordinate plane, label the axes and determine an appropriate scale if you were to graph the speed of a race car over the course of a 2 hour race. Assume that the car will drive at speeds up to 250 miles per hour.
5. Sketch a coordinate plane and graph the linear function f(x) = -300x+250. Understanding x-intercepts:
6. Think about the x-intercept of the graph of the linear function f(x) = 3x+4. What is the y-value of that special point?
7. Find the x-intercept of the graph of the equation represented by y= -15x+60. You may do this algebraically, or with the help of a graphing calculator.
Understanding the zeros of a function:
8. What value of x would make the expression -12x+84 equal to zero? 9. How can a graph help you find the zero of a function?
Understand linear functions:
10. Given the linear equation y= 25x+100: which situation might the equation describe? a. A car drove 25 miles over 100 minutes.
b. A car drove 100 miles at 25 miles per hour.
c. A car was 100 miles away from home when it began driving at 25 miles per hour away from home.
Scale factors:
HO # 3
Crash LandingTask
Names ____________________________________________________ Date_____________________ Problem: An airplane is flying at 36,000 feet directly above Lincoln, Nebraska. Both engines fail and the plane starts going down. A little later the plane is at 28,000 feet directly above Des Moines, Iowa, which is 160 miles from Lincoln. Assuming a constant rate of decent, predict how far from Des Moines the airplane will be when it crashes.
You are an air traffic controller and need to find a safe place for the plane to land. Based on the geographic location, does the plane need to change course to crash in a more ideal spot? How would it have to change? Where will you land it?
1. Draw a picture of the initial situation – on a coordinate plane
2. What is the point where the plane hits the ground?
3. Plot the plane and the possible landing spots to an actual map of the area.
4. Research the landing spots on the map. Where are you going to choose to land?
5. Write your explanation for choosing your landing site.
6. Create “press release” presentation to get information to the public about the crash.
Extension:
7. How does the altitude of possible locations affect this situation?
HO # 4
Crash Landing Task (With Hints)
Names ____________________________________________________
Date_____________________
Problem: An airplane is flying at 36,000 feet directly above Lincoln, Nebraska. Both engines
fail and the plane starts going down. A little later the plane is at 28,000 feet directly above Des
Moines, Iowa, which is 160 miles from Lincoln. Assuming a constant rate of decent, predict how
far from Des Moines the airplane will be when it crashes.
You are an air traffic controller and need to find a safe place for the plane to land. Based on the
geographic location, does the plane need to change course to crash in a more ideal spot? How
would it have to change? Where will you land it?
1. Draw a picture of the initial situation – on a coordinate plane
How far above the city is the plane at each point?
How far apart are the cities?
2. What is the point where the plane hits the ground?
Use slope formula to find the slope of the plane between the 2 cities.
What are the other points around the plane that it could land?
3. Plot the plane and the possible landing spots to an actual map of the area.
Do your measurements match the scale of the map you chose?
Maybe need a side view from the ground, and a top view from the air.
4. Research the landing spots on the map. Where are you going to choose to land?
Think about pros and cons for each possible landing region.
5. Write your explanation for choosing your landing site.
What are the pros and cons?
What is the probability or likelihood of survival?
Are there other things that may have affected this choice?
6. Create “press release” presentation to get information to the public about the crash.
Where was the plane when the engine failed?
Where does the plane land (or crash)?
What was the thought process of the traffic controller, pilot, etc?
What form of media will you use?
HO # 5
Crash Landing Checklist/Rubric
Names ___________________________________________ Date ________________
Possible Earned
Pts
Pts
1. Draw a picture of the initial situation – on a coordinate plane
Points are correctly graphed and labeled
Distances are correctly shown
Axis are labeled appropriately
2. What is the point where the plane hits the ground?
Work shown for slope formula to find decent of plane
Relationship between x-intercept and solution point shown in
work
3. Plot the plane and the possible landing spots to an actual map
of the area.
Map of given area
Points labeled correctly on map
More than one possible solution represented or explored
(several possibilities - extra)
4. Where are you going to choose to land?
Multiple landing options described
Pros and cons listed for each landing option
Final choice clearly stated
5. Write your explanation for choosing your landing site - Paper
Explain your reasoning for your choice
Math results represented in reasoning
Grammar and spelling correct
6. Create “press release” presentation.
Attention grabber
Concise/Understandable
Accurate
Total:
Teacher comments:
Crash Landing: Student self-assessment [Solutions] Slope:
1. The slope of the line describing the plane’s height compared to time in the air is positive.
2. If it goes straight up into the air then the slope of the line describing the helicopter’s height compared to time in the air is undefined.
3. Given two points of data from a linear relation, (-100, 28,050) and (200, 50,000), the slope of the line is 439/6 or approximately 73.17.
Graphing Linear Functions:
4. Sketch a coordinate plane, label the axes and determine an appropriate scale if you were to graph the speed of a race car over the course of a 2 hour race. Assume that the car will drive at speeds up to 250 miles per hour.
5. Sketch a coordinate plane and graph the linear function f(x) = -300x+250. Understanding x-intercepts:
6. Think about the x-intercept of the graph of the linear function f(x) = 3x+4. What is the y-value of that special point? The y-value is zero (0).
7. Find the x-intercept of the graph of the equation represented by y= -15x+60. You may do this algebraically, or with the help of a graphing calculator. The x-intercept is x = 4, or (4, 0)
Understanding the zeros of a function:
8. What value of x would make the expression -12x+84 equal to zero? x = 7
9. How can a graph help you find the zero of a function? It is where the graph crosses the x-axis. Understand linear functions:
10. Given the linear equation y= 25x+100: which situation might the equation describe? a. A car drove 25 miles over 100 minutes.
b. A car drove 100 miles at 25 miles per hour.
c. A car was 100 miles away from home when it began driving at 25 miles per hour away
from home.
