ukanteach 5e lesson plan

UKanTeach 5E Lesson Plan

Author(s): Leslie Rahn Title of Lesson: Free fall and calculating ‘g’

Dates lesson will be taught: 10/7/14 and 10/9/14 Grade level: Grade 11/12 Is this Teach 1 or Teach 2: Teach 1

Lesson Source/References: What did you use to develop this lesson? What lesson planning materials were provided to you? If you adapted a lesson(s) then list the sources here. Lesson developed on my own. -References: 1. http://www.redbullstratos.com/ 2. http://www2.vernier.com/manuals/video_and_logger_pro.pdf

Essential Understanding/Big Ideas: Students will learn about free fall: when the only acceleration acting on a body is the acceleration due to gravity. Students will draw and create position time plots depicting objects in free fall. Students will interpret these plots, and learn how to apply the kinematic equations to data of objects in free fall. Through two exploration activities, students will discover that the acceleration of an object with a large mass (i.e. a skydiver) is the same as an object with a smaller mass (i.e. golf balls, Play Doh). Students will also discover that the acceleration due to gravity is -9.8 m/s/s.

Knowledge Package: See attached hand-written package (included with the reflection write-up).

Objectives: Write specific, measureable objectives using action verbs. Prioritize the list, including at least one higher-order objective. Use the SWBAT form. Students Will Be Able To:

1. Calculate the acceleration due to gravity for various objects (skydiver, golf balls, etc.) using the kinematic equations.

2. Describe and identify the variables in the kinematic equations. 3. Show that the mass of an object does not affect its free fall

acceleration.

Aligning Assessments to Objectives: For each objective identified on the left provide a brief statement that explains how you will measure the objective. What will you ask your students to do to demonstrate understanding? What evidence will you collect?

- Objective 1 and 2: Students will need to use the equations to calculate acceleration using data they are provided with, and with data they collect. If students are able to use the equations, and calculate the acceleration, they will succeed. Questions to ask: Can you identify all of the variables you need for this equation? What information do you have? Do you have enough information to solve for acceleration? Can a different equation be used (i.e. there are 3 kinematic equations)? The worksheet on Day 1 prompts students for a description of the equations – this can be a deliverable to the teacher to show understanding.

- Objective 3: On Day 2, students will explore the p/t plots of two

http://www.redbullstratos.com/

objects (of different mass) using logger pro software and the kinematic equations. Through this activity, students should see that the masses of their objects do not affect the acceleration due to gravity. Questions to ask: What were some accelerations that you calculated for your “heavy” object? What are some accelerations that you calculated for your “light” object? How do these numbers compare? How do these numbers compare to the skydiver’s acceleration? Can you make a conclusion about the effect that mass has on acceleration due to gravity? The students will complete a worksheet on Day 2 with accelerations of two objects. There will be a space for them to conclude something about mass and gravity (acceleration). This will be the deliverable for the teacher to show understanding.

Standards: Include at least one in each category. NGSS Science and Engineering Practice Standard:

S8: Obtain, evaluate and communicate information.

S5: Use mathematics and computational thinking.

HS-PS2-1. Analyze data to support the claim that Newton’s second law…

Planning and carrying out investigations: “plan and conduct an investigation…to produce data to serve as the basis for evidence” Constructing explanations and designing solutions: “…student-generated sources of evidence consistent with scientific ideas, principles and theories.” Science models, laws, mechanisms..: “theories and laws provide explanations in science.”

Common Core Math Practice Standard:

MP.1: Make sense of problems and persevere in solving them.

MP.3: Construct viable arguments and critique the reasoning of others.

Common Core English Language Arts (ELA) Practice Standard:

E3: Obtain, synthesize and report findings clearly and effectively in response to task and purpose.

E6: Use technology and digital media strategically and capably.

Accommodations: -ELL: Students will be working in groups for most of this two-day lesson. Have students discuss their reasoning as they work through the Day 1 worksheet, and be sure that ELL students are teamed up with students who will include them in the discussion and analysis. On Day 2, students will conduct a lab, so ELL students should be working with their team to choose the objects to analyze, participate on the computer analysis and calculations. -Gifted Students: Students who need more of a challenge in this activity may attempt to explain why they chose the equations that they did (or why they didn’t choose a certain equation). Students may also explore what happens when an object is dropped and experiences air resistance (i.e. a feather or a piece of paper). Explain why free fall is only in environments in which air resistance is non-existent.

Safety: Students need to be courteous of other students when conducting the experiment (i.e. being quiet in the halls, respectful of other students, and inclusive of all the members of their team). Students should not throw any of their objects – the only acceleration will be from simply dropping objects (i.e. not from throwing them). All team members need to participate and be included in ALL aspects of the activities.

Equipment/Materials: Day 1: Worksheet for day 1 (20 copies) Teacher PowerPoint presentation, which includes the videos of Felix Baumgartner jump Day 2: Worksheet for day 2 (20 copies) Golf balls, foam balls, ping pong balls, Play Doh (enough for each team of 3-4 to have two objects each) Tablets or other video capturing device (1 for each team, ~5) Logger Pro enabled computers – Advanced prep: Teacher needs to verify that the videos can be uploaded to the computers AND that they videos can be played back on the Logger Pro software. Test everything!

Teach_1_worksheet.

docx

Day_2_worksheet.d

ocx

free_fall.pptx

Engagement: Estimated Time: Day 1: 10-12 minutes; Day 2: 10 minutes

What the teacher does AND how will the teacher direct students: (Directions)

Probing Questions: Critical questions that will connect prior knowledge and create a “Need to know”

Expected Student Responses AND Misconceptions - think like a student to consider student responses INCLUDING misconceptions:

ENGAGE – DAY 1 (50 minute class) Show a short video of Felix Baumgartner’s supersonic fall to Earth. Bring up the PowerPoint slide showing several data points of Felix’s flight. (slide showing only his fall – no numbers yet). Depending on how much they remember, the students may or may not recall much information about the acceleration lab (i.e. a cart rolling down a ramp). In this lab, students found that an accelerating cart will have a curved position/time graph; in other words, the cart’s velocity is not constant.

-What sort of physics is at work here? What are some things that we can calculate or know from this jump? -Why did Felix fall? -But why didn’t he just float next to the balloon? -OK. So the gravity of the Earth is what? What is it doing? -A couple of weeks ago, I saw you doing a lab experiment in which you were studying acceleration. Can someone remind us, what is acceleration? -Alright, we just said that “velocity” describes how fast something is going. Does acceleration mean the same thing? -So acceleration is sort of a combination of how fast something is going (velocity) and how that thing is speeding up? -If I’m driving in my car, and my

-Speed, position, velocity… -How fast he was falling (So, by “how fast”, what do you mean? Give me a physics word to describe “how fast”.) -His velocity. -How long he fell. When he opened his parachute. -Because he jumped out of a balloon. -Because the gravity from Earth pulled him back. -Because he wasn’t in space…there is gravity on Earth -Pulling him back to the Earth. -How fast something is going. -No (I don’t know). -Acceleration is how something is speeding up. How something is going faster. -Yeah. Kind of.

speedometer says “30 mph”, is that velocity or acceleration? -So now I’m merging onto the highway, and I’m changing my “velocity” from 30 mph to 60 mph. What am I doing? -Ah, ok. So if a “unit” of velocity is mph (as an example), what is a unit of acceleration? -Got it. Acceleration is the change in velocity with time. Can you give me some other units of acceleration? What do you see when you do experiments in class? -Alright, let’s go back to Felix’s jump. Do you think he is accelerating? -So when Felix is standing in the door of the balloon capsule, what is his velocity? How fast is he going (downward)? -So if you were standing on the top of a table, just standing there, what is your downward velocity? -So then, Felix jumps out of the capsule. Let’s say he has been falling for 5 seconds. Is his downward velocity zero anymore? -So what you’re telling me is that he went from 0 velocity (let’s just say meters per second), to some non-zero velocity? -So if his velocity is changing with time, what is he doing? -OK. You told me that he is falling

-That’s how fast you’re going… so velocity (speed). -You’re accelerating. -Velocity changing with time… miles per hour per hour? MPH/s? -Meters per second squared, feet per second squared… (let’s agree to use m/s^2… these are SI units). -Yes/no… yes, because he is going faster. No, because he slows down eventually. -He isn’t moving downward. He’s standing still. -His velocity is zero. -Zero. (Right) -No! (This should be obvious at this point. If not, might need more examples. i.e. hold a ball and drop it. Have a student place a book on the edge of the table and then push it off). -Yes. -Yes.

-Well, it turns out that gravity is accelerating Felix. Instead of a gas pedal on a car, we have gravity pulling Felix. And we can calculate this acceleration due to gravity, but we have to know some numbers first. ------------------------------------------------------------- END ENGAGE DAY 1 ------------------------------------------------------------- ENGAGE – DAY 2 (Lab period, or block schedule. If only 50 minute class period, reduce the number of objects the students drop, or make lesson 3 days). Begin by reflecting upon the Felix worksheet In this class period, we will explore the

because of gravity; that gravity is pulling him down to the earth. And you told me that he is accelerating. So what do you think? Are those two things related? ------------------------------------------------------ ------------------------------------------------------ -What did we discover about Felix’s fall last time we met? -What did we find out about his acceleration? Was it constant? -Ah.. What was the maximum acceleration that we found? -OK. What do you think about this acceleration? Why do you think he didn’t accelerate more? -Is it possible to accelerate faster than 9.6 m/s/s? What about faster than 10 m/s/s?

-Accelerating. (Good). -Yes. (or, I don’t know. Or just blank stares…) -Gravity is making him accelerate. ---------------------------------------------------------------- ---------------------------------------------------------------- -He accelerated as he fell. -It too him 60 seconds to get to a certain speed. -No. It increased for a while, and then it decreased. -About -9.6 m/s/s -He opened his parachute (misconception – he doesn’t open his chute for another 4 minutes into the fall). -Don’t know. -Yes/no.

acceleration of various objects.

-Let me ask you another question. You all calculated Felix’s acceleration at these different points. What we haven’t talked about yet is Felix’s shape, weight, height, etc. Do you think these variables had an effect on how fast he fell, and how much he accelerated? -When we left last time, we had a question which was posed: will a bowling ball and a baseball hit the ground at the same time (if they are dropped together)? (The next question is better as an elaborate question) -What about the way he fell? Is there a difference between falling belly first rather than head first?

-Yes. Of course (misconception). If he was heavier, he would fall faster (clarify faster…would he have a different acceleration?) Yes. -Yes/no. (This is what we will explore today). -Yes.. (we’ll get to this later)

Teacher Decision Point Assessment: Include a statement that helps the instructor decide when to move on to the next section of the lesson. Day 1: We need to be sure that the students can relate position, velocity and acceleration. Have the students work in their groups to draw a position time plot, or simply discuss how all three of these things relate. I.e. Acceleration describes how Felix’s velocity changes with time, and velocity describes how his position changes with time. Day 2: Hopefully, students will disagree about the affect mass has on acceleration. At this stage, the teacher wants the students to be discussing mass and acceleration, and debating whether this will change his acceleration. This is enough to move on.

Exploration: Estimated Time: Day 1: 20-22 minutes; Day 2: 60 minutes (including time to set-up experiment)

What the teacher does AND what the teacher will direct students to do: (Directions)

Probing Questions: Critical questions that will guide students to a “Common set of Experiences”


EXPLORATION – DAY 1 (50 minute class) - Go to slide on PowerPoint that shows some numbers of Felix’s fall (i.e. time and position). -Next slide prompts students to begin working on their worksheet – make a position/time plot using the data given. -Have the students work in groups of 3-4. The can use the class whiteboards or the space available on the worksheet to plot Felix’s data. Once the class has completed this (allow about 5 minutes) move to the next slide, which shows your plotted position and time.

-We are given a few numbers here: time and position. What would be a good place to start our analysis of Felix’s fall? -What shape of a p/t plot does an accelerating object have? -(Have the students make a prediction of what the graph will look like before they begin using the data). -What is our zero position in the case of Felix? (Students may be tempted to draw plots that begin at 0,0 but this is not following the data we have) -Follow the prompts on the presentation slide: What does the slope of the line suggest? Is the slope positive or negative? What does this mean? -Our mission is to calculate Felix’s acceleration. What do we need to solve for this?

-We can plot it (or, I don’t know) -We can calculate velocity. -Parabola (curved) -When he is on the ground… -His zero position is where he is standing before he jumps… so way up in the sky is his zero position. -He is accelerating, -The slope is negative, and it is changing (i.e. curved line), negative slope means he was moving down (which is correct… it does not mean that he was moving slow!) -Velocity, time… speed. His mass? (This is a misconception, but write it down and have the students think about it as they work on the problems.) -His position.

Advance the slides to show the Kinematic equations. Students should have seen these equations before, but may not be familiar with them. The equations will also be on the worksheets.

-Last time I visited your classroom, you were plotting position and time and then finding average velocity. But there was a problem with using average velocity to describe all of your data. Remind me what the problem was. What did you have to make smaller to get a better representation of your data? But what is nice about a velocity time plot? If the slope of a position time plot is velocity, what is the slope of a velocity time plot? OK. So if we had smaller time intervals, we could use the velocity time plot to find acceleration. Let’s try another way to find the acceleration. We’re going to try to use math to do this. -Do we have any equations that we can use? -Think about what we are solving for (acceleration), and what equations you already have worked with, or seen. What are these equations? -OK. These are some of the points in

-The slope of the line was not the same for all the data points we had. -The line did not represent all of our data. -The time. -The change in time. -Acceleration. -Yes. -Yes (they may indicate some in the book) -Kinematic equations (equations of motion).

Students will probably need a lot of guidance to get started with deciphering these equations. Here’s what we did:

1. Show the students the kinematic equations. Have them work on the worksheet question which asks them to describe what each term means. Allow ~5 minutes for this.

2. Come together as a class and have the students tell you each of the terms.

3. Remind the students what they are going to try to solve for (acceleration). Ask them which equation they would like to use.

4. Equation 2 is the best choice. The other two equations include the unknown variable “v(t)”, which we just talked about not having.

5. Either in their groups, or with the whole class, have the students write equation 2 so that acceleration is pulled out (i.e. a=(x-x)/t)

6. Once the class agrees with the final form of the equation, have them try to solve for Felix’s acceleration at the first point (i.e. t=5). Then, break into teams to solve for the other accelerations.

**At this point, the teams of four will begin

Felix’s fall. We see there are altitudes and times listed. Do we have enough information to solve for the accelerations at each of these points? -I want you to work in your teams of four. Try to solve for Felix’s acceleration at the points given on the worksheet. As students are working: -Are you noticing a pattern in his acceleration?

-(Misconception): No. All of the equations have position, time and velocity. We know time and position, but we don’t know velocity. -Yes, we do have enough info. (Students should recognize that they have positions, times, and initial velocity. All of these things can give them accelerations at each of Felix’s positions). -Yes, it is increasing. When the groups show the results of their calculation in the EXPLAIN, we can clarify.)

working to solve for the accelerations at the four points on the worksheet. Students should see that Felix’s acceleration increases to about 9.71 m/s/s. ----------------------------------------------------------- ----------------------------------------------------------- EXPLORATION – DAY 2 (90 minute class) The goal of this day is to have the students conduct a lab experiment in which they directly measure the acceleration due to gravity. In the previous lesson, students calculated Felix’s acceleration at various points, and should have found that the acceleration increased up to a certain point. The acceleration did not increase past this amount. In this lab, students will work in teams to find the accelerations of two objects which they will drop from the top of a staircase.

1. In teams of ~3 students, each team will choose 2 objects to drop (i.e. play doh, golf balls, foam balls, etc.)

2. Teams will record the objects being dropped from the top of a staircase using a tablet or other video recording device (TEACHER: Be sure to test that the video can be played in Logger Pro prior to this lesson).

3. One team member will record the drop (be sure the video frames both the top and bottom of the ball’s descent), one team member will drop the objects (individually. i.e. there will be two videos per team), and the

------------------------------------------------------ ------------------------------------------------------ -We want to analyze the descent of the ball (or object). What do we need to make sure of when we record this fall? What are the important parts of the fall?

----------------------------------------------------------------- ----------------------------------------------------------------- -We need to be able to see everything. -We need to see when he drops it and when it hits the ground. -Start, and ending. We need to have the ball

other team member will hold a meter stick in the frame of the video to use as a scale in Logger Pro.

4. Once the videos are recorded, students will upload their videos to a computer, and insert them into Logger Pro.

5. Students will use the video analysis tools in Logger Pro to add data points, set scales and axes, and view the graphs of their objects’ descents.

All of the above steps should be completed in ~35 minutes. If time runs out, have the students analyze only one video per team. Following the guidance on their worksheets, students will then fill in a table of position, time and velocity (all these are given in Logger Pro), and then will try to calculate the acceleration at the different points using the Kinematic equations. Finally, students will plot velocity vs. time in Logger Pro and will use the “tangent” tool to document the accelerations calculated on Logger Pro.

-Do we want to move the camera, or hold it still? (Ask a student to stand up and drop a ball) -How far did that ball fall? -So we need to have some kind of measurement to know exactly how far that ball fell? -So what is the point of holding the meter stick in the frame of the camera? (Allow the students to record their videos and begin working on analyzing them in Logger Pro. It would be very helpful to have some screen shots of the teacher doing certain tasks. These can be either printed out on some paper, or projected using PowerPoint. At the very least, have the instructions projected so the teacher doesn’t have to direct so much). As the students are working on their analysis, some questions to ask:

- What is happening to the accelerations you are calculating?

- What about when you compare the heavy object with the light object?

being dropped and falling and hitting the ground. -Hold it still. -About 2 meters… I don’t know. -We need to measure it. -Yes. -So we know what a meter is… we can use the meter to know some distance.

Teacher Decision Point Assessment: Include a statement that helps the instructor decide when to move on to the next section of the lesson. Day 1: Students need to be able to use and explain all the components of the kinematic equations. They will work on isolating acceleration in one question of their worksheet, and this can be a deliverable to the teacher to check for understanding. Most importantly, the teacher should walk around as the students are working to be sure that they are all able to calculate acceleration. If not, they may need more guidance. Day 2: In this section, we want the students to analyze their videos and fill in the data on their worksheets. Again, be sure they can use and explain the kinematic equations, and that they have calculated acceleration AND found the acceleration using Logger Pro.

Explanation: Estimated Time: Day 1: 12-14 minutes; 15 minutes


Clarifying Questions: Critical questions that will help students “Clarify their Understanding” and introduce information related to the lesson concepts & vocabulary – check for understanding (formative assessment)


EXPLANATION – DAY 1 (50 minute class) Once all the student have calculate their accelerations, have a student write the accelerations on the board (or, each team can make a table of their calculations on their whiteboards: First column, “Time (s)”. Second column, “Altitude (m)”. Third column, “Acceleration (m/s/s)”. ) Begin the discussion of results.

-(Begin by noting any differences between the students’ results) -What is different here? -What do we notice about Felix’s acceleration? -Is his acceleration positive or negative?

-We miscalculated. (Allow the students to recheck their answers. If there is confusion about certain results, calculate the acceleration in questions with the whole class) -It is increasing. -It is going up. -Negative.

The goal of this “Explain” Is to get the students to agree on their acceleration calculations, and see that they found that his acceleration increased to ~9.7 m/s/s. During the next class period, they will observe small objects falling, and will calculate similar accelerations for these small objects. ------------------------------------------------------------ ------------------------------------------------------------ EXPLANATION – Day 2 (90 minute class) In this time, we will discuss the accelerations the students calculated. There should be students who calculated accelerations for heavy objects (i.e. Play Doh) and light objects (i.e. ping pong balls). Also, connect back to Day 1 and the acceleration calculated for Felix.

-Why is his acceleration negative? (Indicate on the position/time plot how his position is moving in the negative y direction – a negative acceleration) -We only have 5 data points here. What do you think his acceleration would look like if we had more data points? -Will his acceleration keep increasing? GO TO DAY 1 ELABORATE ------------------------------------------------------ ------------------------------------------------------ -What were some of your accelerations that you calculated? -What was the greatest acceleration that you calculated? -What about the acceleration of the Play Doh vs. the acceleration of the ping

-Because he’s falling down. -It would keep increasing -He will hit the ground eventually. -Yes/no. He will open his parachute. ----------------------------------------------------------------- ----------------------------------------------------------------- -(Students will tell you several of their points. Hopefully, they will all agree that the highest acceleration was somewhere around -10 m/s/s). -About 10 m/s/s. (9.7 m/s/s)… -They are about the same (they are a little

At this point, explain to the students that this 9.8 m/s/s is the acceleration due to gravity, and that is what free fall is: when the only acceleration acting on an object is the acceleration due to gravity. It is also fun to have the students drop objects in front of the class for a few minutes. This way, they can actually see that the golf ball and the foam ball hit the ground at the same time. To lead into the elaborate for Day 2, ask the students what will happen if you drop a textbook and a piece of paper at the same time.

pong balls? How do those numbers compare? Now think back to Felix’s fall. What was the maximum acceleration that we found for him? So what do you notice for all of these falling objects? And how about our other question that we had? Does the acceleration increase much past 9.7 m/s/s? -Will the paper hit the ground at the same time as the book? -Why not?

different, but not much). -9.7 m/s/s. -They all have about the same acceleration. -Mass has no effect. -No. They are all about the same. -No. -Because of air resistance. Because it will float down.

Teacher Decision Point Assessment: Include a statement that helps the instructor decide when to move on to the next section of the lesson. Day 1: We are leaving day 1 a little open ended – we want the students to have calculated all the accelerations for Felix’s descent, but we are leaving with two questions: 1. If we had more data points, do you think Felix’s acceleration would increase much past what we have calculated? Why/why not? 2. If Felix were much heavier, what would his acceleration look like? Have a brief discussion with the students about these two points. What we want is a bit of disagreement: some students thinking that a heavier object will fall faster, and some thinking that they won’t. Day 2: Students should see that, no matter the mass of an object, the acceleration due to gravity will be the same.

Elaboration: Estimated Time: Day 1: <5 minutes; 5 minutes (if time allows)


Probing Questions: Critical questions that will help students “Extend or Apply” their newly acquired concepts/skills in new situations


ELABORATION – Day 1 (50 minute class) There will probably not be much time left in the class, but we want to leave the students thinking about 2 questions:

1. Will Felix’s acceleration continue to increase if we had many more data points?

2. If Felix’s mass were much different (i.e. if he was 150 pounds heavier), how would that affect his acceleration?

ELABORATION – Day 2 By now, there will not be much time remaining in the class period, but we can begin to discuss the effects of air resistance. Namely, if a text book and a piece of paper were dropped at the same time, what would happen. Have the students discuss this… some students in my class had a good feel for air resistance.

-What would his acceleration look like if Felix was 150 lbs heavier? -Do you believe this? Do you think a heavy object and a light object will hit the ground at the same time? These are the questions we are going to try to answer during the next class. Let’s keep these in mind.

-He will accelerate much faster… he will be going more than 9.7 m/s/s. -Nothing. Because mass does not affect acceleration. (One student in my class said that if you drop a bowling ball and a baseball at the same time, they would hit the ground at the same time. Other students disagreed) -Yes/no.

Teacher Decision Point Assessment: Include a statement that helps the instructor decide when to move on to the next section of the lesson. Were students able to calculate g? Do they see that mass has no effect on the acceleration? What will happen with a piece of paper?

Evaluation: Estimated Time: ____5 minutes each day Critical questions that ask students to demonstrate their understanding of the lesson’s performance objectives.

Formative Assessment Day 1: Explain the types of formative assessment you will use to monitor student learning within this lesson (e.g., observations, bell ringers, exit slips). Embed student copies of these assessments, if applicable. Embed copies of the rubric or answer keys you will use to evaluate responses, if applicable. Identify the formative assessment you will give to all students at the end of Day 1. This has to be something you can collect and bring back with you. -I’ve written all the formative assessments in the body of this lesson plan. All of my questions were formative and the worksheets were formative. Formative Assessment Day 2: Explain the types of formative assessment you will use to monitor student learning within this lesson (e.g., observations, surveys, white boards). Embed student copies of these assessments, if applicable. Embed copies of the rubric or answer keys you will use to evaluate responses, if applicable. -Again, I’ve written all of this down in the lesson plan above.

Summative Assessment (at the end of Day 2): Explain the summative assessment you will use to provide evidence that students have met the objectives of the two-day lesson. Embed a student copy of this final assessment. Embed a copy of the rubric or answer key you will use to evaluate responses. The summative assessment was meant to be the worksheet for the final day – students were to fill in the data tables and then write a statement about what effect mass has on a falling object and what the acceleration due to gravity is.

Extension activities/Back up plans: Include a brief summary of some extension activities you could draw upon if you have extra time and/or you are not able to execute this lesson as planned (e.g., technology issues prohibit you from doing so). -When the tablets decide that they don’t want to participate well, use the video camera which is recording the lesson.

ukanteach 5e lesson plan

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