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Transformation 2013 Design Challenge 5E Lesson
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What I Can Derive From Velocity Mrs. West Transformation 2013 Calculus
Grade Level:
Challenge Abstract:
In this Design Challenge, students will design a roller coaster for Six Flags Fiesta Texas that relies solely on the force of gravity to move the cars. The students will specifically investigate derivatives and their roles in instantaneous velocities.
Meeting the Needs of STEM Education through Design Challenge
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Design Challenge Title: What I Can Derive From Velocity TEKS addressed: 2.1 Develop the concepts of the derivative; 2.2 Have an understanding of the derivative at a point; 2.1.b Interpret a derivative as an instantaneous rate of change. 2.3.a Determine the corresponding characteristics of the graphs of f and f ′. Non-targeted Newton’s laws of motion, initial force, gravitational force, friction, distance/rate/time relationship, coordinate plane graphing, flowcharting
Engage Activity Approximate time: 15 Min Show the video clips (see the Resources section below for the web links to these clips). Ask the students to study the clips and be prepared as groups to present to the class all of the similarities and differences they can find. Have them specifically target math/science concepts. Present findings. Introduce Design Challenge
Engage Activity Products/Artifacts
Groups will prepare non-linguistic representations (see Resources section below for additional information regarding non-linguistic representations) to show similarities and differences in the three video clips.
Materials/Equipment: – Computer – Projector – Internet connection Resources: Non-Linguistic Representations: http://www.netc.org/focus/strategies/nonl.php Video Clips: http://vids.myspace.com/index.cfm?fuseaction=vids.channel&ChannelID=4653975 http://www.guzer.com/videos/longest_motorcycle_jump.php http://www.motorcyclevideos.co.nz/view_video.php?viewkey=295f8076b1c5722a46aa http://www.break.com/index/snowmobiler_wipes_out_hard.html http://www.break.com/index/time_to_reconsider_career_choice.html
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Explore Activity 1 Approximate time: Provide a table full of different objects (i.e. bowling ball, marble, ping-pong ball, toy car, balloon, and dodecahedron). The students are to study the objects as they relate to movement. Provide them with planes of different surface materials (items with varying textures that can be used as “ramps”) which can be placed at different angles. Document the things which affect movement. 1. How does “pushing harder” affect movement? 2. How does surface angle affect movement? 3. How does surface texture affect movement 4. Does the shape of the object affect movement? 5. Predict what object at what elevation on what surface will move the fastest. 6. Experiment to verify prediction. Students are to answer the questions and make their predictions in their journals. They need to describe the results of their experiments. Summarize what they discovered about objects and motion. Explore I Activity Products/Artifacts
Journal Entry
Materials/Equipment: – Journal – “objects & ramps” from above…ie. – Bowling ball – Ping-pong ball – Marble – Balloon – Toy car – Dodecahedron – Scrap wood – Book – Small square piece of cardboard – Rough sand paper to drape over a smooth surface, etc. Resources: None
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Explain Activity Approximate time: 12 min Conduct a class discussion to share observations and things learned through the discovery time. Introduce vocabulary as the discussion takes place (i.e. Initial force, gravitational force, friction, Newton’s laws of motion, distance/rate/time relationship, and velocity). Each group will produce a Frayer model-type representation of the terms. Groups discuss where these terms were relevant in their first journal entries
Explain Activity Products/Artifacts
Frayer models for vocabulary terms.
Materials/Equipment: – Journals – Frayer Model: Resources: http://www.longwood.edu/staff/jonescd/projects/educ530/aboxley/graphicorg/fraym.htm
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Explore Activity 2 Approximate time: 40 min As a class, students are to use the available materials to create a track layout that contains a “loop de loop”. The marble must stay on the track and complete the entire course. After the first attempt, the students are to document successes and failures. If the marble does not complete the course, they are to make modification hypotheses to correct the problem. After the group comes to a consensus on the best plan, they are to make the appropriate adjustments to the track. Continue this process until they are successful. Use a stopwatch to record the time it takes for the marble to “run” the course. Explore 2 Activity Products/Artifacts
Students are to summarize the design process in their journals. Their summary should contain the problems encountered and adjustments performed. They should include the data of the track length and the time recorded on the stopwatch. Each student is to construct a flowchart in their journals to describe the process of designing a successful track.
Materials/Equipment: – Pipe insulation – Pre-cut wooden dowels – Marbles – Duct Tape – Toothpicks – Craft sticks – Stopwatch – Journals Resources: http://library.thinkquest.org/2745/data/ke.htm http://library.thinkquest.org/2745/data/loops.htm http://www.brainpop.com/science/energy/kineticenergy/ http://www.physicsclassroom.com/Class/energy/U5L1a.html http://hyperphysics.phy-astr.gsu.edu/hbase/ke.html http://hyperphysics.phy-astr.gsu.edu/hbase/pegrav.html#pe
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Explain Activity Approximate time: 25 min Conduct a group discussion of their efforts that encourages students to pull in the vocabulary from their journals. Demonstrate how to use the length of the track and the time measurement to determine the average velocity of the marble. Have each group draw a replica of the track. Each group should 1. Decide if the marble’s rate was constant. 2. Determine when the rate of the marble changed. 3. Mark the replica to point out where the marble was accelerating and decelerating. 4. Mark the 3 places where the marble was moving the fastest and the slowest. 5. Predict what would happen if the circumference of the loop was made smaller or larger. Introduce the concept of instantaneous velocity and describe how it relates to average velocity.
Explain Activity Products/Artifacts
Each group will submit their track replica with markings and average velocity calculations. Each student is to write a summary of their findings in their journals. They should include their predictions about changes in the size of the loop
Materials/Equipment: – Journals – Large paper for each group to complete the replica – Markers – Ruler Resources: None
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Explore Activity 3 Approximate time: 40 min Using the track from the previous activity, mark the track in one inch increments. Position a stopwatch (one that measures in tenths of seconds with a large display) so that it can be video recorded along with the marble running the course. The students should practice releasing the marble and timing the event. Make a video of the race event with the stopwatch and track increments visible. As you show the video, pause several times as the marble is moving. Record the position of the marble and the time displayed on the stopwatch. Record the data. Continue to view and pause collecting time/position data.
Explore 2 Activity Products/Artifacts
Each student is to construct a time/position graph using the data. Each student makes a journal entry to predict how the time/position graph can be use to discuss velocity as a rate of change with respect to time. 1. Can you look at the graph and determine when the velocity is most constant? When is the velocity changing
Materials/Equipment: – Digital video recorder – Computer – Projector – Rulers – Markers – Stopwatch – Graph paper – Journals Resources: None
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Explain Activity Approximate time: 45 min Demonstrate the concept of instantaneous velocity by analyzing the time/position graph. Draw the line tangent to the curve at a few key points on the graph. Discuss the slope of the tangent line as a rate of change of distance and time. Have students discuss the positive/negative slopes as well as their relative steepness and how that relates to the changes in velocity at those positions on the track. Introduce the derivative as a way to calculate instantaneous velocity. Compare the graph of a function and its derivative. Consider the tangent line of the velocity graph. Discuss the definition of acceleration. Explore Activity Products/Artifacts
Provide guided and independent practice using the tangent line/ 1st derivative to calculate instantaneous velocity. Students compose a journal entry to summarize their findings. They are to predict how this same idea could be used to find acceleration.
Materials/Equipment: – Journals – Text (or use suggested link) for guided/independent practice Resources: http://www.themathpage.com/aCalc/motion.htm http://cda.morris.umn.edu/~mcquarrb/CI/HW/2.1.pdf – Any Calculus Text
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Elaborate Activity Approximate time: 120 min
Given 25 feet of track, students are to design a track that contains the maximum possible number of loops. Their layout must be designed to fit in the classroom, and the marble must be able to negotiate and complete the track. Elaborate Activity Products/Artifacts
1. Each group will turn in a track proposal which includes a diagram of the track along with a written justification for how and why their design meets the specification. 2. Each group will have design brief. 3. Groups will then build their track and run their marbles. Photograph the track layout. 4. Each student will journal the results and come up with suggested modifications for track failures. 5. Groups will review suggestions and modify their tracks. 6. Repeat the test/modify phase until successful or until time runs out. 7. Each student is to write a 1 to 2 page paper of the design process. The summary should include how they used the knowledge from the explore activities to be successful and they should be able to explain the math principles they learned/used. Completed track design that meets the specifications.
Materials/Equipment: – Pipe insulation (enough for each group to have 25 feet of track) – Wooden dowels – Saw – Marbles – Duct Tape – Toothpicks – Craft sticks – Journals – Camera Resources: None
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Evaluate Approximate time: Homework due 2 to 3 days after tracks completed Students will turn in a portfolio that includes the following: 1. Diagram with justification 2. Design brief 3. Pictures of each prototype 4. Journal entries for failed attempts 5. Design Paper Each group will demonstrate their design.
Evaluate Activity Products/Artifacts
Materials/Equipment: – The completed track layouts – Students bring their portfolios Resources: None
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Diagram with justification Design brief Pictures of each prototype Journal entries for failed attempts Design Paper Final track layout