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Thank you for the feedback during yesterday's discussion. I'm excited to try this!
3622:12721
ShareUsing Your Marbles: Making Energy Work for You
Summary
Alisa Hilfinger, Westborough High School, Westborough MA
This activity is based on the common experiment of running a marble down a ramp to do work on a cup. Many different versions of this type of activity exist and will pop up under a general Google search. Students will be able to see the relationship between mass and work done on the cup and/or ramp height and work done on the cup.
9th Grade introductory Physical Science class
Learning Goals
The first goal is to give the students an activity in which to anchor our discussion of potential and kinetic energy. Students must "mess about*" enough with the materials to create a model to observe the effect of ramp height on the distance the cup moves AND/OR the effect of marble mass on the distance the cup moves. A group discussion after the activity will inform all the students of what they each discovered. The students will be given specific supplies and specific instructions on what they needs to accomplish (the marble must do work on the cup without the student pushing on the marble) but they will determine HOW to make it happen.
The second goal is to lay the framework for an inquiry based experiment where the students gets to ask "What would happen if...." about the model they designed. Each student will be allowed to pursue his/her own question, gather data, analyze the data and come to a conclusion.
The third goal is for students to practice communicating through pictures. Each student will record observations about their model in whatever format they choose, but will be required to present an illustration of their model and of their observations.
Context for Use
I am using this to introduce the idea of energy. In this format, it is an introductory activity to provide a common experience to which all students can refer during discussion of mechanical energy. After we understand gravitational potential and kinetic energy, including calculations, we will revisit the activity to give students an opportunity to re-design the materials into a new model and take measurements for basic calculations of gravitational potential and kinetic energy.
The initial activity is planned for around 30 minutes, followed by 20-25 additional minutes of discussion and modeling of what the students discovered. Note: my intention is to not use the words "energy," "potential" or "kinetic" unless the students initiate it, but to experience the cause/effect relationship between the ramp height, marble mass and cup movement.
Background
Prior to this activity, we will have finished a unit on Work. The depth of coverage will depend on the level of students. The notion of work done on the cup is the bridge into the topic of energy.
WORK
When a force is applied to an object and the force causes the object to move a distance parallel to the direction of the force, work is done. Assuming a constant force (in both magnitude and direction), work can be calculated as W = F d, where F is the component of the force parallel to the direction of motion and d is the resulting displacement of the object. The units are Newton meters, called joules (J). This is the extent to which the 9th grade students in an introductory 1/2 year physics course explore work.
Work can, however, be taken to greater depth and be calculated as W = F d cos θ where F is the magnitude of the constant force, d is the magnitude of the object's displacement, and θ is the angle between the directions of the force and the displacement, therefore allowing non-horizontal forces to be explored. The total work done on an object should also consider all the forces experienced by the object. For an example of a box on the floor being pushed by a person, the forces would include the force of the push, the friction of the floor, the weight of the box (gravity) and the normal force of the floor on the box.
The goal in introducing the activity to my students in the context of them having to make the marble do work on the cup is that they can make the connection of the marble applying a force to the cup causing the cup to move a distance. From here the students will, hopefully, start exploring how much work can be done depending on how high the ramp is and the mass of the marble.
ENERGY
At the most basic level, energy is the ability to do work. Energy is work that hasn't been done yet. One analogy is that if you have $5 in your hand and $5 in the bank, each amount of money has the same value and are measured in the same units. They can buy the same items. The $5 in your hand that you give to the cashier is similar to "work" because it is the use of the money in the moment. The $5 in the bank is similar to "energy" in that it is money you can use later. You possess it, but you aren't using it at the moment to purchase anything. If the conditions are right (such as you have access to your bank), you can use the money later. Similarly, if the conditions are right, you can use energy to do work.
In confining our lesson to conservative forces (a more in-depth analysis of work-energy principles will include non-conservative forces as well) we will specifically discuss gravitational potential energy and (translational) kinetic energy.
Kinetic energy (KE) is the energy of motion and the motion of an object will increase with an increase in mass and also with an increase in velocity. If the mass is doubled, the kinetic energy is doubled and if the velocity is doubled, the kinetic energy is increased by four.
Kinetic energy = ½ mv2 where m is the mass of the object (kg) and v is the velocity (m/s). The units are equal to a Newton meter, or joule.
Gravitational Potential Energy (GPE) is the energy of position based on the vertical height of an object relative to some reference level below. Gravitational potential energy of an object on Earth is the result of the object's weight and its height above the ground (or whatever is defined as the ground level).
Gravitational potential energy = mgh where m is the mass (kg), g is acceleration due to gravity on earth (9.8m/s/s) and h (m) is height above the reference level. The units are equal to a Newton meter, or joule.
The total mechanical energy of a system (with the constraint of no nonconservative forces present) is equal to total kinetic energy plus total gravitational potential energy. The sum of KE and GPE is constant if only conservative forces are acting on it, thus meeting the conditions for the principle of Conservation of Mechanical Energy. This principle states that the total mechanical energy is always the same, no matter how it is divided between KE and GPE. Accordingly, if KE decreases, GPE much increase and if GPE decreases then KE must increase.
The transition from KE to GPE and back is very abstract and difficult to follow, even with examples of objects moving under the influence of gravity without friction. A hypothetical rollercoaster, wrecking ball, marble down a ramp, skateboarder in a half-pipe are all examples that can be discussed. Having experienced one of these models not in the context of KE and GPE, however, allows the students to intuitively grasp the relationship between height and speed. The leap to labeling the position and speed of the marble in terms of GPE and KE makes the abstract concept easier to grasp and provides shared context for discussion.
Based on the mathematical notion that work and energy are measured in the same units, and on the observations that as position changes, speed changes therefore affecting the amount of work done on the cup, provides evidence to help solidify the idea that the joules of mechanical energy that an object has based on its position and motion is equal to the total amount of work that can be done by that object.
Description and Teaching Materials
In-Class Activities
Students will be allowed to work individually or grouped with no more than 3 in a group and given access to the materials with explicit instructions to use these tools to create a system in which the marble can do work on the cup under its own abilities (not pushed!). Students will be encouraged to spread out and set up their systems on the floor. It is anticipated that the systems will look like this:Ruler, marble, cup figure (Microsoft Word 20kB Jun24 10)
Ideally, the leading edge of the ruler will be pushed against the far lip of the cup so the ball is not rolling along the floor before it hits the cup. This is not a detail that will be pointed out during the "messing about" phase but a detail that will be discussed in designing the inquiry-based lab at the end of the unit.
Once students reach a point of having the ruler act as a ramp that the ball can roll down to push the cup, I will circulate and possibly ask questions to try and encourage the students to see if they can get a controlled response from the cup (make it go farther or not as far).
Each student will create a drawing of his/her system illustrating the observations on the work done by the marble on the cup.
As a class, we will discuss our observations.
At Home Assignments (followed by more in-class activities!)
Create a table with 3 columns: "What I Know", "What I want to Know", "What I Learned" and complete the table based on our activity.
This will be the jumping-off point for discussion on Day 2. We will review our K/W/L tables and each student will draft at least one "What would happen if..." question based on the information from his/her "What I want to Know" column. We will create a master list of of questions and post them on the wall for the duration of the unit.
This will segue into our (several class days) discussion of energy including learning how to calculate GPE, KE and ME. Students will then evaluate the marble/ruler/cup system in the context of energy and students will gather data to try and answer their individual "What would happen if..." questions.
Materials
plastic ball bearings and steel ball bearings of equal volume (called marbles throughout this lesson)cups with cut-out in lip to accommodate the width of the ruler and be tall enough to accommodate the ruler on an angle with clearance for the largest marble
rulers with groove down the entire length that will act as ramps for rolling the marbles down to the cup
books/blocks/notebooks; items to stack for students to use to create the ramp at different heights
Standards
Introductory Physics, High School: Learning Standards for a Full First-Year Course
Content Standards
2. Conservation of Energy and Momentum
Central Concept: The laws of conservation of energy and momentum provide alternate approaches to predict and describe the movement of objects.
2.2 Interpret and provide examples of how energy can be converted from gravitational potential energy to kinetic energy and vice verse.
(Massachusetts Science and Technology/Engineering Curriculum Framework, October 2006 (pg 74))
Teaching Notes and Tips
This activity is designed as the introduction to energy BEFORE using the terminology. The intention is to provide a common experience for all students to anchor future discussions because I find that not all students can describe a personal experience on a roller coaster or skate board or a wrecking ball (!). Ideally, this activity will be revisited after the students gain knowledge on the topic so that they can individualize their experiment to make a better model and gather data for simple calculations.
I am especially curious to see what the responses are for the homework assignment of posing a "what would happen if" question. Similar to David Hawkins approach in "Messing About in Science" when he created index cards with suggestions for relationships to explore for the pendulum, I will accumulate the students' "what if" questions into a master list and let them choose what to explore. A student may decide to stick with his/her original question or be interested in another. I'm hoping that exploring a personalized question provide motivation for the students to follow through with designing a model that will allow for data collection and mathematical analysis.
This activity could easily be scaled up or scaled down for different age groups. Students don't have to gather mathematical data for calculations and can evaluate each "what would happen if" exploration on the basis of visual data. Alternatively, students can layer their levels of understanding by exploring additional variables such as changing the shape of the cup or the surface the cup runs across. There are also many different kinds of measurements that can be made, some of which might include GPE of the marble, velocity of the marble at various points along its path to calculate the KE of the marble at those points, force applied to the cup, and accounting for non conservative forces.
I am especially curious to see what the responses are for the homework assignment of posing a "what would happen if" question. Similar to David Hawkins approach in "Messing About in Science" when he created index cards with suggestions for relationships to explore for the pendulum, I will accumulate the students' "what if" questions into a master list and let them choose what to explore. A student may decide to stick with his/her original question or be interested in another. I'm hoping that exploring a personalized question provide motivation for the students to follow through with designing a model that will allow for data collection and mathematical analysis.
This activity could easily be scaled up or scaled down for different age groups. Students don't have to gather mathematical data for calculations and can evaluate each "what would happen if" exploration on the basis of visual data. Alternatively, students can layer their levels of understanding by exploring additional variables such as changing the shape of the cup or the surface the cup runs across. There are also many different kinds of measurements that can be made, some of which might include GPE of the marble, velocity of the marble at various points along its path to calculate the KE of the marble at those points, force applied to the cup, and accounting for non conservative forces.
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Assessment
The initial assignment of illustrating the model will be evaluated based on general completeness such as including each of the materials in the picture and if observations have been included.
After the (several day) discussion of energy, including calculations, students will revisit their original illustrations of the model and label the path of the marble according to amount and type of energy the marble has for those spots including (but not limited to) 100% GPE, 100%KE, 50%KE, 50%GPE. This will be assessed based on accuracy.
Students will choose a "What would happen if" question for the culminating experiment and record the initial question, pose a hypothesis, write a procedure, collect data, analyze data with calculations and draw a conclusion in a formalized lab report. Students will hand in one per student.
After the (several day) discussion of energy, including calculations, students will revisit their original illustrations of the model and label the path of the marble according to amount and type of energy the marble has for those spots including (but not limited to) 100% GPE, 100%KE, 50%KE, 50%GPE. This will be assessed based on accuracy.
Students will choose a "What would happen if" question for the culminating experiment and record the initial question, pose a hypothesis, write a procedure, collect data, analyze data with calculations and draw a conclusion in a formalized lab report. Students will hand in one per student.
References and Resources
Giancoli,D., 2005, Physics Principles with Applications, 6th Edition. Pearson Prentice Hall
Trefal, J., et. al., 2006 Physical Science, McDougall Littell
Hawkins, David., 1974, "Messing About In Science," The Informed Vision, Essays on Learning and Human Nature. Agathon Press
Trefal, J., et. al., 2006 Physical Science, McDougall Littell
Hawkins, David., 1974, "Messing About In Science," The Informed Vision, Essays on Learning and Human Nature. Agathon Press
Using Your Marbles: Making Energy Work for You --Discussion
James Tanton
Jun, 2010
G'Day Alisa:
Here are some rambling thoughts from a mathematician who really knows nothing.
1. (NOT RELATED TO YOUR ACTIVITY): One thing that drove me nuts as a kid in physics class was the origin of formulas. I was under the impression that quantities described in physics class (work, force, energy, etc) really "exist" and we're finding formulas for them. It took me a long while to realise that, like mathematics, we're creating concepts and creating formulas to match those concepts we created. For example, I have no idea what "force" is, but I am willing to say that F = ma is a *definition* of force, not so much an equation for it. Also, I have an intuitive feel of what "work" is: Suppose I push a box. If it is twice as massive, it is twice as hard to push. If I push it twice as far, I feel I have done double the "work." So work is some concept that I feel should exist that is directly proportional to mass and directly proportional to distance. I can make up most any formula I want, but easy is always good so let's *define* work to be: W = F*d. SO this is what phsycists do over and over again. They observe things in the universe, develop an intuitive feel for what seems to be happening, and then try to create formulas for concepts that seem to mimic what we feel. We could just as well define work to be 7*F*d or 15*F^2*d^3 (but that doesn't match our intuition as well). We always like the simplest approach.
The notion of potential energy drives me nuts. You refered to a "reference level" so if I hold a brick up in the air and ask how much potential energy does it have, I would answer mgh where h is the height above the floor. But if I told you I was actually on the second floor and want to use a different value for h, then the value of my potential energy has changed? So what does this mean for the law of conservation of energy? (My personal definition of potential energy is: that which makes the law of conservation of energy work!)
Anyway, I guess I am just saying that it might be a good pre-activity for kids (or a mental experiment at least) to derive a formula for "work" by starting with the idea of pushing a box and asking "on which variables would you like it to depend? - that is, changing what changes the amount of work you do?" and then, "what's the simplest formula you can write that relates those quantities in the way you feel is right?" I bet you'll get W = fd from this.
[BTW: The formula for KE is really from calculus. There seems to be this notion of "momentum" in our intuition and the simplest formula one might imagine for it is mv. Then "energy" is a concept that somehow affects momentum of an object. A change of energy gives a new momentum for an object, and we're in calculus land. It seems the simplest thing to define is KE is that which fits the equation:
d(KE)/dv = momentum.
Integrating gives, KE = (1/2)m v^2. It seems to me, in my naive understanding of things, that formulas in physics are really based on capturing intuive notions in the simplest mathematical way possible. We're creating them.]
2. I love the idea of doing this activity before you even use the standard vocab for any of this. That, to me, is so important. I teach many of my math classes this way, letting the kids make up names for things which happens only when there is a need for something to be named. (Why name something if there is no need yet for it being named?) We then have a discussion on "what we called it versus what the rest of the world calls it" (e.g. the word "powers" versus the word "logarithms.")
This is a great activity. Despite all my ramblings above I really have nothing deep to say about what you've put together here. I am very eager to hear how it actually goes with a group of kids. You'll see then, I am sure, what adjustments and refinements and reorderings are needed.
Grand stuff!
- J
Here are some rambling thoughts from a mathematician who really knows nothing.
1. (NOT RELATED TO YOUR ACTIVITY): One thing that drove me nuts as a kid in physics class was the origin of formulas. I was under the impression that quantities described in physics class (work, force, energy, etc) really "exist" and we're finding formulas for them. It took me a long while to realise that, like mathematics, we're creating concepts and creating formulas to match those concepts we created. For example, I have no idea what "force" is, but I am willing to say that F = ma is a *definition* of force, not so much an equation for it. Also, I have an intuitive feel of what "work" is: Suppose I push a box. If it is twice as massive, it is twice as hard to push. If I push it twice as far, I feel I have done double the "work." So work is some concept that I feel should exist that is directly proportional to mass and directly proportional to distance. I can make up most any formula I want, but easy is always good so let's *define* work to be: W = F*d. SO this is what phsycists do over and over again. They observe things in the universe, develop an intuitive feel for what seems to be happening, and then try to create formulas for concepts that seem to mimic what we feel. We could just as well define work to be 7*F*d or 15*F^2*d^3 (but that doesn't match our intuition as well). We always like the simplest approach.
The notion of potential energy drives me nuts. You refered to a "reference level" so if I hold a brick up in the air and ask how much potential energy does it have, I would answer mgh where h is the height above the floor. But if I told you I was actually on the second floor and want to use a different value for h, then the value of my potential energy has changed? So what does this mean for the law of conservation of energy? (My personal definition of potential energy is: that which makes the law of conservation of energy work!)
Anyway, I guess I am just saying that it might be a good pre-activity for kids (or a mental experiment at least) to derive a formula for "work" by starting with the idea of pushing a box and asking "on which variables would you like it to depend? - that is, changing what changes the amount of work you do?" and then, "what's the simplest formula you can write that relates those quantities in the way you feel is right?" I bet you'll get W = fd from this.
[BTW: The formula for KE is really from calculus. There seems to be this notion of "momentum" in our intuition and the simplest formula one might imagine for it is mv. Then "energy" is a concept that somehow affects momentum of an object. A change of energy gives a new momentum for an object, and we're in calculus land. It seems the simplest thing to define is KE is that which fits the equation:
d(KE)/dv = momentum.
Integrating gives, KE = (1/2)m v^2. It seems to me, in my naive understanding of things, that formulas in physics are really based on capturing intuive notions in the simplest mathematical way possible. We're creating them.]
2. I love the idea of doing this activity before you even use the standard vocab for any of this. That, to me, is so important. I teach many of my math classes this way, letting the kids make up names for things which happens only when there is a need for something to be named. (Why name something if there is no need yet for it being named?) We then have a discussion on "what we called it versus what the rest of the world calls it" (e.g. the word "powers" versus the word "logarithms.")
This is a great activity. Despite all my ramblings above I really have nothing deep to say about what you've put together here. I am very eager to hear how it actually goes with a group of kids. You'll see then, I am sure, what adjustments and refinements and reorderings are needed.
Grand stuff!
- J
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Lindy Elkins-Tanton
Jun, 2010
Hey, Alisa -
I like the idea of taking a standard activity and putting on a twist to make it more educational and interesting. The resulting activity, I suspect, will be useful to a lot of teachers.
The analogy troubles me since it implies there are temporal aspects to work and energy -- energy can also be right here, right now, too, just like the work you suggest. And it can be hard to store and transport energy, so not many energy banks function well.
A really good activity, seems like it's ready to go!
all the best -
Lindy
I like the idea of taking a standard activity and putting on a twist to make it more educational and interesting. The resulting activity, I suspect, will be useful to a lot of teachers.
The analogy troubles me since it implies there are temporal aspects to work and energy -- energy can also be right here, right now, too, just like the work you suggest. And it can be hard to store and transport energy, so not many energy banks function well.
A really good activity, seems like it's ready to go!
all the best -
Lindy
3622:12781
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Ellen Doris
Jun, 2010
Hello Alisa,
I think providing a common experience that everyone in the class can refer to is important. It’s inclusive, it grounds discussion in actual observations, and it also makes it easy to return to the materials/phenomena as questions or disagreements arise. The particular experience you’ve chosen is simple and well suited to your goals.
It would be helpful to me (and perhaps to teachers who consider using your activity) if the specific concepts you want kids to understand were highlighted succinctly under “Learning Goals.”
Regarding your homework assignment: Are the things that students will write in the “What I know” column their ramp and marbles observations? If so, “What I noticed” or "Observations" might capture that, and distinguish it from “What I Learned.” (You might be after something else here, though.)
I’ll be very interested to hear how Hawkins’ structure (messing about/multiply programmed, guided investigations/lecturing-discussing-theorizing) materializes in your classroom and whether you feel it serves you and the students well.
Ellen
I think providing a common experience that everyone in the class can refer to is important. It’s inclusive, it grounds discussion in actual observations, and it also makes it easy to return to the materials/phenomena as questions or disagreements arise. The particular experience you’ve chosen is simple and well suited to your goals.
It would be helpful to me (and perhaps to teachers who consider using your activity) if the specific concepts you want kids to understand were highlighted succinctly under “Learning Goals.”
Regarding your homework assignment: Are the things that students will write in the “What I know” column their ramp and marbles observations? If so, “What I noticed” or "Observations" might capture that, and distinguish it from “What I Learned.” (You might be after something else here, though.)
I’ll be very interested to hear how Hawkins’ structure (messing about/multiply programmed, guided investigations/lecturing-discussing-theorizing) materializes in your classroom and whether you feel it serves you and the students well.
Ellen
3622:12872
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