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Reifying vector components

I asked the following question when discussing 2D kinematics in Physics 1 this semester:

A ball is kicked with an initial speed of 32 m/s at an angle 40o above the horizontal.

a) What are the x and y components of the ball’s velocity 1 second and 3 seconds after being kicked?

b) How high will the ball go?

c) 60 m in front of where the ball is kicked is a 12 m tall wall. Will the ball make it over the wall?

d) For the same initial angle, what is the minimum initial speed required for the ball to make it over the wall?

Ignore the fact that 32 m/s is somewhat unrealistic as a speed for a kicked ball. What makes part d) more difficult than part c)? Take a moment to answer to yourself.

My initial thought when writing this question was that parts d) and c) differed primarily in their algebraic complexity. In part d), students need to solve for time in terms of v0 and then plug that algebraic expression for time into a second equation for vertical position. I was wrong. The algebra is nothing compared to the difference in how vector components need to be conceptualized.

In part c), students are given numerical values for v0 and theta and they can use v0*cos(theta) and v0*sin(theta) to calculate x and y components of the initial velocity. A student can successfully complete part c) thinking of v0*cos(theta) and v0*sin(theta) as nothing more than algorithms. A student who thinks of these expressions as procedures for computing x and y components with no physical meaning on their own can be successful on part c).

In part d), a student must recognize v0*cos(theta) and v0*sin(theta) as entities that have physical meanings in and of themselves. By this I mean that a student who knows to use v0*cos(theta) to calculate v0,x does not necessarily know that v0*cos(theta) can be used in equations in place of v0,x. In order to solve part d), a student must see v0*cos(theta) and v0*sin(theta) as entities rather than just procedures. These expressions need to be seen as every bit as physical as v0,x and v0,y.

This point was driven home to me when a student who had solved part c) asked for help on part d). I said (unhelpfully) that you follow the same process as for part c). The student wrote 60=v0,x*t and said ‘but I don’t know vo,x‘. I asked if the student could relate v0,x to the initial speed and the angle and he wrote v0,x=v0*cos(theta). I then pointed at this expression and said that we want to replace vo,x in his first equation with v0*cos(theta) because then we can solve for t in terms of v0. The student’s response was, ‘Oh, I didn’t realize we could use this (v0*cos(theta)) to, like, represent the x velocity.’ What an eye opener for me as an instructor!

This is a nice example of something Sfard calls reification. The Closes introduced me to Sfard and reification a while back, but I need to have another read and think about the implication for vector components.


An unexpected error with waves

On the first waves exam I included a question where I gave students y vs. x plots of a wave at t=0 and at t=0.1 s. I then asked the following questions:

a) Determine the wavelength, period, and wave velocity.

b) Write the wave function y(x, t) that describes this wave.

Students are reading values off a graph so there is some wiggle room in the values they obtain for each quantity. Without thinking about it, I used a wave with an amplitude of 2.4 m and a wave speed of 2 m/s. While grading the exams, I came across several students with amplitudes in y(x, t) that were slightly off, but I chalked this up to being in a hurry and not reading the graphs carefully.

Then I came across a student who explicitly labeled the amplitude of y(x, t) as the wave speed. This error never occurred to me and because I chose such similar values for amplitude and wave speed I can’t determine if other students are making this error or if they are just sloppy graph readers. Sigh… something to be more careful about next time I write questions.

Inside vs. Outside Views of Comedy and Teaching

I read Tina Fey’s book Bossy Pants over the winter break and at one point she lists her favorite lines from each of the writers on 30 Rock. What struck me was that while I find 30 Rock to be extremely funny, none of the lines she highlighted stand out to me as particularly funny. I’ve noticed this when I listen to interviews or podcasts with comedians as well. Very often, the things a professional comedian finds very funny are not the same things I find very funny. Professional comedians seem to find it extremely funny and wonderful when someone defies expectations or plays with timing and delivery in an unexpected way. From my perspective, I often don’t notice or am not automatically delighted by such things.

This brings me to teaching and PER. It occurs to me that if two people, one a seasoned teacher and one not, or one a PER person and one not, both observe the same class with the same instructor, they will notice and delight in different instructional moves. If after the class both observers are asked what the instructor did that they thought was particularly good, they are likely to give different answers.

This seems obvious as I write this, but it occurred to me on the drive to work today that I don’t always keep this in mind during instructor observations. There’s a tendency to see an observation as one sided – the observer takes notes and then tells you what they think you did well and where you could improve. My last observation went well and I received high marks, but I should go into my next observation debriefing with my own notes about what I think I did well and where I think I struggled. This way I can open up the discussion and talk about *why* each of us thinks certain instructional moves were or were not successful and how they do or do not align with my instructional goals. This also gives me more appreciation for the importance of observational protocols like the RTOP or UTOP and for the importance of the follow-up discussion after the observation.

Acceleration with mixed units

On a recent Physics 1 homework I asked a question about boats in which the velocities were given in mph meaning the acceleration value (what students calculated) came out in miles/hr^2. I told students that miles/hr^2 values are difficult to think about so they should convert their acceleration into values in miles per hour per minute and miles per minute squared. I then asked students to explain in words what their acceleration value in miles per hour per minute means.

Most students are giving incorrect explanations for what the acceleration means, but that is not unusual. Acceleration is a difficult concept to wrap your head around. However, it seems useful to me to ask students to think about acceleration in mixed units. In this case there are explicitly two units of time that students need to decide how to work into their explanation. The seems pedagogically more helpful than asking students to explain the meaning of an acceleration of 5 m/s^2. We’ll need to discuss the question in class tomorrow, and I need to think about whether its best to introduce the question in class or in homework but I think there’s good potential here.

Aside: I spent some lecture time when first introducing acceleration encouraging students to always think about acceleration units as meters per second per second rather than meters per second squared. Unfortunately, I haven’t continued to enforce or utilize this convention in class discussions.

How do I engineer this experience?

I’d like to do a better job of noticing small, day-to-day successes in my classes. I look at strengths and weaknesses on a unit by unit or topic by topic level but there are so many tiny minute to minute things that also contribute to the success of a lesson. One example comes from our first lab on waves.

The first lab is very qualitative and starts with students making various waves on Slinkies. The Slinkies are long, so we have groups spread through the classroom and the hallways outside. As I was wandering from group to group, I noticed one group in which a student was vigorously shaking the end of the Slinky while the other end lay on the ground about 4 feet in front of him and about 3 feet from his partner. After a moment, I realized that the student was trying to get the free end of the Slinky to slide towards his partner so no one would have to get up to grab the free end.

I watched for a few seconds and when the student started to give up I commented that the student should know why that won’t work. The student paused for a moment and then excitedly said, “Oh yeah, I should do a longitudinal wave!” and started shaking the Slinky longitudinally. I waited a few more seconds and then asked, “What’s the defining characteristic of a wave?” This was a question on that week’s reading quiz that every student had answered prior to class. The student thought and his partner replied, “Waves transport energy not matter.” To which the student nodded, “Oh, so it will never work.”

Success! A dry-sounding definition becomes meaningful and physical for these students.

What is sticking with me and bothering me is how this experience came about. The students were “goofing around” and I just happened to be walking by when it happened. If I hadn’t been walking by just then and hadn’t made a comment, I doubt the students would have pondered why their plan didn’t work. If I had written this question into the lab activity, I doubt the students would have internalized the concept in the same way. Can I engineer this moment to happen again in the future or is this just a ‘right place at the right time’ sort of thing?

In an attempt to extend the experience to the rest of the class, I described the scenario in a Hw problem and asked students if they should a) make a transverse wave, b) make a longitudinal wave, or c) give up and have someone get up to grab the free end. About half the class chose c) citing that waves transport energy not matter and about half the class chose b) saying that they just need a longitudinal wave with a 3 foot amplitude. In either case, I am doubtful that the question led to an ah-ha moment for anyone.

Other than a healthy respect for “goofing off” in lab, is there something more productive I can take away from this experience?

Waves LGs: Sp14

These learning goals for Unit 1 of Physics 3 are very similar to the LGs I used last semester. Primarily I tried to add a little more specifics, particularly to the LGs about the derivation, intensity vs. distance, and standing waves. I added a specific LG talking about resonance and clarified and expanded the LGs relating specifically to electromagnetic waves.

I use “Explain” too often in these LGs. I need to sit down with a list of Bloom’s verbs and try to fix this. I also want to sit down and write a few bullet points for each LGs about what specifically it would look like for a student to demonstrate the LG. This should also help improve my verbs. Plus, not every LG can be assessed on an exam (some for practical reasons and others simply due to limited time) so writing these bullet points will force me to plan ahead for LGs that don’t fit into an exam assessment.

Content Learning Goals

Upon successful completion of this unit the student will be able to…

1. Explain what does and does not propagate as a wave travels from point A to point B and illustrate this idea by talking about a specific type of mechanical wave. Explain the meaning of wave velocity in light of these ideas.

2. Explain what amplitude, frequency, angular frequency, period, wavelength, angular wave number, phase constant, and wave velocity mean physically and relate these quantities mathematically. The student should know what these quantities mean for both transverse and longitudinal waves and should be able to determine their values given a graph or an equation for a wave function.

3. Write the wave function for a wave traveling in a given direction, at a given speed, with a given amplitude and wavelength. The student can also show that this wave function satisfies the linear wave equation (note that this requires knowing the linear wave equation).

4. Write the wave function for a sinusoidal wave given its graphs (y vs. x and y vs. t) or vice versa.

5. Derive mathematically, along with an accompanying explanation and diagram, one of the following (student’s choice): the linear wave equation for a transverse wave on a string, the speed of a transverse wave on a string, or the speed of a sound wave traveling through a gas. The student should explain and justify any small angle approximations, changes in reference frame, or derivatives used in the derivation.

6. Determine the speed of a wave on a string or the speed of sound through a gas based on properties of the medium. The student can use these results to compare the speeds of waves in different mediums or to compare the speeds of two different waves in the same medium.

7. Define intensity and offer a geometric explanation for how it varies with distance from the wave source in 1D, 2D, and 3D. The student can relate amplitude to intensity and can calculate the amplitude and intensity at different distances from the wave source for water waves, light waves, or sound waves.

8. Explain how decibels relate to sound intensity, and why we use decibels to describe sound intensity. The student can also explain how decibels and intensity relate to loudness and what change in decibels or change in intensity corresponds to a sound “twice as loud”.

9. Explain in words what causes the Doppler effect and what changes (apparent wavelength or apparent speed) when the source is moving compared to when the observer is moving. The student can relate frequencies and speeds mathematically for when the source and/or the observer is moving.

10. Explain the idea of interference or superposition in words and diagrams and given examples of how this phenomenon can be observed in sound waves, water waves, light waves, and waves on a string.

11. Explain the role interference or superposition plays in creating a standing wave and describe the behavior (and physical meaning) of nodes and anti-nodes in a standing wave. Student can also describe the boundary conditions for a standing wave on a string with fixed or free ends and for a standing sound wave in a pipe with open or closed ends as being nodes or anti-nodes. For sound waves, the student can do this for pressure or particle displacement.

12. Sketch a standing wave corresponding to the fundamental frequency or higher harmonics. Student can also calculate the fundamental frequency, harmonic frequencies, and associated wavelengths for a standing wave on a string or in an air-filled pipe.

13. Apply (12) to musical instruments like guitar, flute, or blowing across a Coke bottle to explain how to produce sounds with higher and lower frequencies.

14. Describe the phenomenon of beats and their cause in words. Student can explain the difference between the conditions that create standing waves and the conditions that create beats.

15. Explain the phenomenon of resonance in driven waves and the relationship between resonance, interference, and standing waves. Use these ideas to explain the results of shaking a string with fixed boundary conditions at different frequencies and of striking a tuning fork over tubes of varying lengths.

16. Explain in words the idea of a Fourier series and why it is useful.

17. Explain in words the idea of displacement current and its associated magnetic field.

18. Explain how electromagnetic waves differ from mechanical waves and the role that accelerating charges play in creating EM waves. Student can also mathematically relate the amplitude and phase of the electric and magnetic fields in an EM wave.


Dimensions, Vectors, and Kinematics LGs: Sp14

I’m teaching calculus-based Physics 1 for the first time this semester. Actually, Calc I is a co-requisit not a prerequisite for this class so I’ll show students some calculus but they won’t do much themselves until Physics 2. Anyway, here is my current set of learning goals for Unit 1.

Content Learning Goals

Upon successful completion of this unit the student will be able to…

1. Recognize the power of 10 associated with the prefixes: nano, micro, milli, centi, kilo, mega, giga. The student can move fluently between numerical powers of 10 (i.e. 10-3) and prefixes (i.e. milli) in conversation and in written problems.

2. Recall the S.I. units for length, time, speed, acceleration, and mass. The student also knows how various algebraic operations affect units and can use this knowledge to keep track of units while solving problems.

3. Recall approximate (order of magnitude) values in meters and kilograms for: the height of an adult, the mass of an adult, the length of a football field, the length of a mile, and the mass of a car. The student can also recall approximate values in meters per second for: a person running, a car driving, a plane flying. The student can use these values to solve estimation problems and to assess the reasonableness of his or her answers to various problems.

4. Determine when dimensions must match and when they need not match when combining terms. The student can explain the meaning of the phrase ‘dimensional analysis’ and can use dimensional analysis to assess the appropriateness of new equations and to construct approximate equations for order of magnitude problems.

5. Explain the difference between velocity and acceleration and can give examples of motion in which: one quantity is zero and the other is non-zero, one quantity is positive and the other is negative, each quantity points in a different direction. The student can also explain the meaning of the phrase ‘rate of change’.

6. Explain the difference between a vector and a scalar and classify common quantities such as position, distance, displacement, speed, velocity, acceleration, mass, and density as scalars or vectors.

7. Differentiate between instantaneous and average values for speed, velocity, and acceleration. The student can calculate instantaneous and/or average quantities given: a table of values, a graph, a description of motion in words, or a kinematic equation. The student can also explain under what types of motion the average and instantaneous value of a quantity will be the same.

8. Recall equations v = v0 + at, v2 = v02 + 2ad, <v> = (v + v0)/2, and x = x0 + v0t + 1/2at2. The student can explain under what conditions these equations are valid and can use them to setup and solve kinematics problems.

9. Translate between different ways of representing motion: description in words, x vs. t or v vs. t graphs, equations, and motion maps. Given any representation, the student can construct any of the other representations. The student can also explain when each representation might be useful.

10. Define ‘free fall’ and explain what free fall implies and does not imply about the values of kinematic quantities v0, v, and a.

11. Explain the meaning of the magnitude of a vector and the meaning of symbols \vec{A}, |\vec{A}|, and for a general vector. The student can also calculate the magnitude of a one-, two-, or three dimensional vector.

12. Draw a right triangle for any 2D vector and use SohCahToa to determine the components of the vector.

13. Add or subtract two or more vectors both graphically (for 1D or 2D) and algebraically (for 1D, 2D, or 3D) and give the results in cartesian or polar coordinates. The student should be able to do this regardless of whether the original vectors are given in cartesian or polar coordinates. The student can also multiple a vector by a scalar both graphically and algebraically.

14. Explain the meaning of unit vector and recall how unit vectors are denoted. The student can recall how i-hat, j-hat, and k-hat  relate to the x, y, and z axes. Given a vector , the student can represent this vector in either (Ax, Ay, Az) notation or in Ax i-hat + Ay j-hat + Az k-hat notation.

15. Determine the value and direction of velocity and acceleration for an object thrown straight up when the object is: in the hand being thrown, halfway to the top, at the very top, halfway down, in the hand being caught.

16. Solve 2D kinematics problems by separating the motion into horizontal and vertical components and applying our basic kinematics equations (LG 8) to each component individually.

17. Define ‘projectile motion’ and solve for maximum height, horizontal distance, and time of flight for a projectile. The student can also calculate horizontal or vertical position, displacement, velocity, and acceleration at any moment during the projectile’s flight.

I’ve not written General Skills or Experiential learning goals for this unit.