Zen and the art of predicting voltage across series resistors

I have a circuit in which a 14 V battery is connected to a 15 ohm and a 25 ohm resistor in series.  What's the current through and the voltage across each resistor?

In my honors-level classes, I teach a mathematical solution using the VIR chart.  They calculate the equivalent resistance of 40 ohms; use ohm's law on the total circuit to get a current of 0.35 A; recognize that series resistors each take that same 0.35 A current; then multiply across the rows of the chart with ohm's law to get 5.3 V and 8.7 V across the resistors.

In conceptual, though, we don't use a calculator, and I want to minimize (not eliminate) calculation, anyway.  So we approach this problem slightly differently.  

Take a look at the worksheet we've used in laboratory to learn how to deal with series resistors.  Students must justify their answers to each question thoroughly.

1. Which resistor carries a larger current through it?

We start with a not-so-subtle reminder of the rule that each resistor carries the same current. 

2. Which resistor takes a larger voltage across it?

Now, I insist on an equation justification with ohm's law.  V = IR; I is the same for each because series resistors carry the same current.  By the equation, then, the larger 25 ohm resistor takes the larger voltage.

3. What is the equivalent resistance of the circuit?

Fact: the equivalent resistance of series resistors is the sum of the individual resistances.  Just add 'em up to get 40 ohms.

4. Calculate the current through the circuit.

Only now do we do a calculation.  We've learned that ohm's law applies for the total voltage and resistance in a circuit.  So, we simply say that I = V/R, with V = 14 V and R = 40 ohms.  This makes the current 14/40 amps. Yes, I allow that as the answer -- we aren't using calculators, and I don't want to mess with issues of significant figures and decimals.  Look, go ahead and haul me before the Klingon Death Tribunal for my sins.  I'm looking forward several years.  When they see circuits in a senior honors class, they'll be able to call this 400 milliamps.  For now, I'm happy that they know which values to plug in to ohm's law.

5. Estimate the voltage across and current through each resistor in the chart below.  You may use fractions for current, but not for voltage.  Answers without units earn no credit.  

Here's the Zen.  Not calculating current through each resistor -- that's the same 14/40 amps through each.  The Zen is the estimate of voltage.  I'm NOT teaching them to multiply the resistance of each resistor by the current.  Nor am I teaching them to proportionalize the voltage according to the resistance.  Nope.  I'm just saying "estimate."  

All I am looking for is an answer that fits the facts they've already stated:  the voltage across each must add to 14 V, and the 25 ohm resistor must take greater voltage.  Some students will guess 2 V and 12 V; some will guess 6 V and 8 V.  I don't care.

6.  Now, set up the circuit, and MEASURE the voltage across each resistor, and across the battery.  Record your results here.

And now the Zen must be reconciled with reality.  They make the voltage measurements, and see how the voltage is distributed:  in this case, about 5 V and 9 V.  Some students got the estimate right -- they get candy.  I praise everyone else for a "good guess;" we look to see whether their guess was high or low for the 25 ohm resistor.  

I don't teach anything, still; instead, I hand out a new sheet, with different resistors.  They fill it in again, all the way from the beginning, steps 1-6.

As you may or may not suspect, by the second or third time most students are getting pretty dang close to the right voltages.  Some folks discover the proportionality rule for themselves.  Others just recognize that "close" resistors demand "close" voltages, and "far apart" resistors demand disparate voltages.  

To me, this process is teaching good physics.  I've taught the calculations for series resistors for ages, and I've been repeatedly frustrated by students who can make calculations well but can't answer simple conceptual questions like "which takes the bigger voltage."  And as well by students who frustrate themselves because they predicted 8.25 V but only measured 8.22 V.

By the second day of filling out these sheets and measuring voltages, these freshmen are getting almost bored with the process.  That's the sign I'm looking for.  When I start to see faces saying "Gawd, not again with the voltage question..." that's when I know it's time to move on.  I give the faster guys a sheet with three, not two, resistors; then, after a multiple choice quiz, we move on to resistors in parallel.  

I'll teach parallel resistors the exact same way.

GCJ

Circuits: first day introduction for 9th grade

In my 9th grade class, I'm expecting students to deal with simple questions and calculations with series and parallel resistors, though not with a single circuit combining both series and parallel resistors.  We are teaching series resistors first, for a week; we'll deal with parallel resistors later.

My approach to circuits is not pure modeling, but somewhat close.  I don't discuss the subatomic nature of current, or any sort of analogy for current and voltage in a circuit.  Rather, I define terms based on what we can see and measure:  "Voltage is provided by a battery," for example.  My goal is to get, as quickly as possible, to experimental work in which students build circuits and measure voltages.

My colleague Alex Tisch and I made this helpful handout for the first day.  You might want to print it out, or load it in an adjacent window, as you read this post.

I write the basic definitions and units on the board, and students copy these in preparation for a quiz the next day.  The definitions I use are in this file.  We quickly get to two demonstrations.  

First, I help the class make a prediction for what a graph of current vs. voltage at constant resistance should look like.  Since resistance is constant, V=IR suggests we should get a straight line graph.  

Next, we actually do the experiment.  I set my "decade box" variable resistor to 50,000 ohms, and vary the voltage from 2-20 V.  My ammeter measures the current, usually getting in the hundreds of microamps.  Notice that I've already set up reasonable axes in a blank graph on the handout.  I have each student in turn come to the front of the room, adjust the voltage, read the voltmeter, and read the ammeter.  I write a table of values on the board, and everyone graphs at his seat.  Sure enough, 7-8 data form a pretty clear line.  

Okay, then... what if we hold the voltage constant and change the resistance?  We predict... V=IR says that as resistance increases, current decreases.  We decide as a class that a 1/x graph is more likely than a linear upward-sloping graph*.  The experiment and the students' graphs verify the prediction, this time setting the voltage at 10 V and changing the resistance between 1 thousand ohms and 100 thousand ohms.

Though I don't use those terms... I sketch a 1/x graph and a y=x graph, and without naming them just ask:  which sketch is more likely?

On day two, I actually hand out my definitions for series resistors.  We read them and practice using them with an actual circuit with series resistors.  By day three, I'm ready to give them each a breadboard and have them predict and measure voltages across series resistors.  

Using this approach, my students will be slower than most in figuring out how circuits work.  A more traditionally taught class* will build some competence more quickly.  But I'm convinced that after a few days of predicting, connecting, and measuring their own circuits, my class will catch up with and perhaps pass by traditionally-taught classes.  And they'll pick up parallel resistors more easily.  Why?  Because they're not calculating dispassionately.  They're making predictions for their personal circuit, predictions that they personally will test!  It's amazing how much more they care when mother nature is sitting there, ready to prove them right or wrong.

* i.e. one in which you do practice calculations repeatedly in class and for homework

GCJ

[P.S. If I get to it, I'll make a future post explaining how I have everyone calculate the voltage across each resistor without calculating.  Yes, I know, it's Zen.]

Doppler demo -- with an onion bag

photo credit Graham McBride

Doppler effect problems in textbooks often involve trains and trumpeters.  That's all great, but I want to demonstrate the Doppler effect in my classroom, in which I have neither train nor trumpet.  Furthermore, when I've actually used a trumpeter playing a note out the window of my car, it's been hard to hear the effect -- I can't go much faster on the road here than 20 mph, and I'm not confident that my student trumpeter can maintain a constant pitch while turned sideways in the front seat.*  I need something better.

* Band veterans probably are ready to interject here the ol' wisecrack about the definition of the "minor second" being two high school trumpet players attempting to play the same note.  

My favorite Doppler effect demonstration involves twirling a speaker in a circle.  You get the speaker to play a constant frequency, attach it to a string... then when the speaker is briefly traveling toward the listeners, the pitch increases noticeably, especially compared with the decreased pitch that happens less than a second later.  The "wahh-wahh-wahh"   two-toned pitch is easy for everyone to hear.

The problem I've always had with that demo is the physical setup.  I attach a rather bulky frequency generator to a cannibalized speaker using alligator clips.  If I'm not extra careful, the alligator clips fail, and the speaker goes flying.

Nowadays, I use my iphone or ipad as a frequency generator with either the "freqgen" app or the "tone generator" app.  My question this morning was, how do I whirl my phone in a reasonably high-speed circle without the risk of breaking the phone?  Tying string to the iphone wasn't getting me anywhere.

My Chinese-teaching colleague Scott Navitsky gave me the key suggestion:  use an onion bag instead of string!  I asked our dining services for an onion bag, and I thank Jim Robertson and Aimee Carver for providing me with one.  I told the app to play a 600 Hz note, stuck the phone in the bag, twirled... and the warbling frequency was apparent to everyone nearby.

GCJ

Questions about AP Physics 1 and 2

Georgia teacher Mark DiBois sends in the questions that virtually everyone in the country who teaches AP Physics B is dying to have answered:

Just read the e-mail about the sweeping changes coming in AP Physics B.  Now its going to be Physics 1 and Physics 2.

Yup, this has been in the works for a while, now.  The College Board has set a date:  May 2014 will be the last AP Physics B exam, and the new exams will be released in May 2015.

Can you fill me in on what the premise behind this is?

Take a look at this post from 2010 and this post from 2011.  Then look at the official College Board home page for AP Physics B and click on the redesign link.

Then take a look at the "curriculum framework", which has now been released publicly.

I read the curriculum and It didn't make a lot of sense to me.

I know.  That's the glaring weakness in the College Board's well-intentioned, and generally quite successful, effort to take Physics B ever farther away from the show-me-your-algebra-skills content of the 1980s and into the era of explaining physics with words.  Once the education PhDs got themselves involved, all hope of a one-page topic listing vanished.  In the effort to be transparent and specific about the exam, the College Board instead has written an impenetrable document, one that must be parsed as carefully as the infield fly rule.*

*Which, as TBS announcers showed during the Braves-Cardinals game, is incomprehensible even to purported baseball experts.  Don't get me started on that one.  Suffice it to say that Sam Holbrook got the call exactly right, and the TBS booth should be disbanded on grounds of competence.

The best bet is to learn over the next few years how to use a low-pass filter to eliminate the buzzwords, the eduspeak, and the myriad "the student can engage in...." fluff.

Take a look at the example questions on page 131 of the curriculum framework.  Read them.  Be sure you can solve them.  Don't bother with the "targeted learning objectives" -- just read the questions, and look for patterns indicating how this test will be different from Physics B.  For example:

• See the ranking task: not "what is the power dissipated by the 110-ohm resistor," but "rank the energy dissipated by each resistor in a fixed time."
• It doesn't ask, "calculate the initial speed of the car," but rather "can the speed of the car be determined, and why or why not?"
• Not "at what time on the v-t graph is the cart at rest," but "describe in words the motion of the marble represented in the graphs above."
• Note the request in the free response question to "justify your answer qualitatively, with no equations or calculations."  [my emphasis.]

What's going to come out of all the blabber is a course which demands that students be able to express a clear understanding of physics topics using WORDS.  

Now, that doesn't mean you should stop teaching calculational physics!  My own perspective is that for AP-level students, calculation is a step toward serious conceptual understanding expressed verbally.  If they can explain correctly, they can calculate, too; but if they can't calculate, they can't explain, either.  You're going to have to spend the time to go beyond just complicated problem solving, and into making the students explain why they solved problems the way they did.  Good physics B teachers are already doing this, but are pressed for time.  How nice that the new exams aren't as broad as Physics B.

What will be the basic topics for each?

Look on page 152 for Physics 1, and on page 160 for Physics 2.  This gives the "concepts at a glance."  It's still way too much information for a quick overview; but it's a start.  Perhaps eventually I will try to digest the topics down to a one-page cheat sheet.  You need that cheat sheet -- as long as you understand the level of deep verbal reasoning required in each topic.

If each course is a year long... do you have to take both courses to get 1 college credit?

Ach, the old credit question.  It's a reasonable question, but it's as answerable as "what is the sound of one hand clapping."  My answer: who knows.  Don't ever believe anything you hear about AP credit or placement policies unless it is in a personal communication with the registrar of the college you are considering.  

The best advice for our students: take AP Physics 1 as a first-time physics course.  That's how it's intended.  Do well on the exam.  Then, if you have a year of high school left, take AP Physics 2 or AP physics C.  Don't worry about college credit until you're at college.    

The best advice for teachers is to place your top first-year physics students into AP Physics 1.  Then offer either AP Physics 2 or AP Physics C as a second-year course, depending on your interest, and on whether your students are sophisticated mathematically.  (Physics 2 does not require calculus or any math higher than Algebra I / geometry; Physics C requires fluency at college-level calculus.)

These are good courses, courses that I encourage everyone to try before dishing out the boilerplate "Aarrgh, change!" complaints.  Every day, we ask our students to adjust to new ways of thinking in order to tackle new and scary physics problems.  I think it only fair that we make the effort to jump into a new course that requires us to change the emphasis of our teaching a bit.  Right?

GCJ

Using a lookup table for a conceptual physics lab

In conceptual physics, I want to do an experiment with a 60 Hz frequency generator and waves on a string.  The setup is shown in the picture to the right: the hanging mass is varied, varying the tension in the string and thus the wave speed and the wavelength.  We move the generator left and right until the standing waves are clear; then we measure the wavelength with a ruler.

I want to plot wave speed vs. wavelength, so that the slope of the straight-line graph will be the 60 Hz frequency.  

Problem is, I don't have an instrument to measure wave speed on the string.  In AP physics, I'd just show the students the equation 

and let them figure out the wave speed for themselves.

Well, this is 9th grade conceptual physics. Most of my students either have not completed algebra 1; most wouldn't know a square root if it bit them on the arse.*  I can not expect my class to be able to plug into this formula.  But I still need them to be able to graph a wave speed, knowing only the mass of the hanging mass.

*That happened to me once.

One thought I had was to create a quick app to make the calculation:  On an iphone or ipad, it could ask "What's the hanging mass?"  Then, using the linear mass density value I measured for the string before class, I could program the app* to spit out "the wave speed is 3000 cm/s."  Yes, I know I could do something like this in excel or on wolfram alpha, perhaps, but anything beyond a mass input in grams followed by a speed output in cm/s is too complicated for me.

*That is, if I knew how to program ios apps.  Hey, now, if I had access to a 1985 version of applesoft basic, I'd pwn all of ya in a programming contest.  And I'd have that "app" ready in five minutes.

Without the ability to make the program I want, I realized that I could go all 1940s and just create a lookup table.  Excel will do the calculation... in fact, I learned how to get excel to round the speeds to two significant figures.  So I put mass values from 5 g to 300 g in one column.  I made excel use the equation above to calculate the wave speed in units of cm/s.  

Then I just printed the two columns.  I'll hand this out to each lab group.  I think it's totally reasonable to expect freshmen to use this table to relate the hanging mass to the wave speed... then to graph wave speed on the vertical, and the measured wavelength on the horizontal.

GCJ