Good morning. I’ve been a high school physics teacher for more than 2 decades
and I require you, my students, to “Show All Your Work!” I often get asked why. Today,
I answer that question. ♪ Flipping Physics ♪
I need to start with a little background about me and my teaching. During roughly my first 5
years of teaching I was mostly trying to tread water as a teacher, father, husband, and son.
During that time my wife and I got married, purchased land general contracted our new house,
and had 2 children. One of those 2 kids was born with many physical issues which
required more than 200 doctor’s visits and 3 surgeries before their second birthday.
I started teaching AP Physics B. My father was diagnosed with cancer and died 19 months later.
And I introduced a new second year physics course of both AP Physics C: Mechanics,
and Electricity and Magnetism. Again, all of this during my first 5 years of teaching.
During that time, I did not really require you, my students, to “Show All Your Work!”
and honestly, it was not good. At that point I made a conscious decision to help my students
learn better and the first step in that process was requiring you to “Show All Your Work!”
Y’all fought back against it and continue to this day to question the validity of this approach,
however, I can tell you that you, my students, understand what you are learning much better when
you “Show All Your Work!” Let’s talk about why. Please. Do.
One of the first assignments I have you, my students, do, is a lab which utilizes
very little of what you have learned in the class and honestly, you probably could have
done the lab a couple of years ago. The whole purpose of this first lab is to make sure you
can “Show All Your Work!” The first question on the lab is to take measurements to determine the
density of a steel sphere in kilograms per meters cubed. Here are my solutions to this question.
I start by measuring the mass of the steel sphere using an electronic balance. It is 68 grams. I
write that down as one of my known values. And then I measure the diameter of the steel sphere
using digital calipers. It is 25.3 millimeters. I also write that down as one of my known values.
Now, I know I am going to need the radius and not the diameter of the steel sphere, so I write down
an equation which relates radius and diameter. Radius equals half the diameter and I substitute
25.3 millimeters in for the diameter and get 12.65 millimeters for the radius of the steel sphere.
I now write down what we are solving for: density equals question mark.
Next, I know we are going to need the volume of the steel sphere, so I write down the equation
for the volume of a sphere, it equals four thirds pi times radius cubed. Then I substitute in 12.65
millimeters for the radius and get 8,479.3 millimeters cubed for the volume of the steel
sphere. [show calculator] Now, my calculator gives me a lot more digits than I have written
down. I usually write down 2 or 3 more digits beyond the number of significant figures in the
problem and then use the answer button on the calculator when I do future calculations. And
then round to the correct number of sig figs when I get to an answer. We are solving for density,
so I write down the equation for density: density equals mass divided by volume.
We have both those numbers, so I can substitute them in. [show calculator] 68 grams divided by
8,479.3 millimeters cubed gives us 8.0195 times 10 to the negative 3 grams per millimeter cubed.
However, the question asks for the density of the steel sphere in kilograms per meter cubed, so we
need to do a conversion. I’ve been talking for quite a while. Billy, please do that conversion.
Absolutely, uh. First, we need to convert from grams to kilograms. We know 1000 grams equals
1 kilogram. So, multiply the density by 1 kilogram divided by 1000 grams to convert
from grams to kilograms. Now we need to convert from cubic millimeters to cubic meters. We know
1 meter equals 1000 millimeters. So, multiply by 1000 millimeters over 1 meter, and we get …
It’s cubic millimeters though and not just millimeters, right?
Right. So, just cube it and we get 1000 cubic millimeters over 1 cubic meter. And we get …
I do not think that is correct. … Oh, right, we need to cube everything in
the parenthesis. It’s 1000 cubed, millimeters cubed, 1 cubed, and meters cubed. Thanks.
You’re welcome. And we get 8,019.5
kilograms per cubic meter. And we round to 2 sig figs because the mass only had 2 sig figs,
and we get 8.0 times 10 to the 3rd kilograms per cubic meter for the density of this steel sphere.
Thank you both. A few things to point out about my solution. I listed the values I measured as
my knowns. I wrote down the equation relating radius to diameter and used it to determine the
radius. It is not unusual for students to measure the diameter, immediately divide by two, and then
only write down the radius. The problem with that is that we did not measure the radius, we measured
the diameter. Our known values can only be things we are actually able to measure. I wrote down
what we are solving for: density equals question mark. Please, always identify what you are solving
for. It really helps because sometimes, believe it or now, we forget what we are solving for.
I wrote down the equation for the volume of a sphere, substituted in a number for radius, and
wrote down the number, with 5 significant figures, for the volume of the sphere, including units.
I wrote down the equation for density, substituted in numbers for mass and volume, solved for the
density in grams per cubic millimeter, converted to kilograms per cubic meter, and then rounded
to 2 significant figures because the least number of sig figs from our knowns was the mass at 2 sig
figs. Bo, this is a typed-out reproduction of your solution. Please tell me what you are missing.
Sure, uh. There is so much missing.
I know. (Hey! No snickering.
We have to learn from one another, and scornful laughter is not going to help.)
Oh, yeah. Sorry. Sorry Bo.
It’s okay. The truth is, we are all going to make mistakes.
We learn from our failures. If we do not feel comfortable failing, we do not learn.
So, please, be better than that. Bo, please critique your own solution.
I did not identify my known values. They are just numbers. I need to say mass equals 0.0868
kilograms, for example. Actually, the electronic balance measured the mass of the sphere in grams
not kilograms. So, I need to write down the known mass in grams not kilograms.
I did the thing you mentioned where I never wrote down the diameter of the sphere, only the radius,
even though we did not measure the radius. I did not write down any equations. I should have
written down the equations for volume and density before I used them. I rounded the
volume of the sphere even though that’s not an answer and I did not give units for the volume.
I did not show the conversion from grams per cubic millimeter to kilograms per cubic meter
and I did not show any units on those numbers. Which means I have 8.0 times ten to the negative
3 equals 8.0 times ten to the positive 3 and those clearly are not actually equal.
I did not show the unrounded density before I rounded it for my answer.
I guess I missed a couple of things. Well done Bo, and don’t worry, you are not
alone. Before students can turn in their solutions to the first lab in my class, I require that they
show their solutions to me so we can discuss how well they did with the whole “Show All Your Work!”
thing. With a solution like the one you just gave me; it is not unusual for students not to be able
to describe what they did. When I point to parts of your solution and ask, “What did you do here?”,
with solutions like this, students typically struggle to be able to tell me. Alternatively,
with a solution where all the work is shown, you do not have to remember because everything
is right there, you can just read it! Bobby, let’s look at one step in your solution to this problem.
Bobby, tell me what you see. Uh, I solved for the radius.
(Did you?) Well, kind of. …
(But, following all the equal signs from left to right, this statement says the diameter
equals 12.65 millimeters. Is that correct?) No, okay, I think I solved for the radius
in my head and then divided the diameter by 2 to get the radius. That equal sign in the
middle is not actually correct. I should have put an arrow in the middle to show
that you can use diameter equals 2 times the radius to get radius equals half the diameter
and then solved for the radius. That makes sense. Very nice Bobby. The essential difference is that,
when equations on both sides of the equal sign are not equal to each other, an arrow symbol is
used to separate steps instead of an equal sign. The arrow indicates that one equation can be
rearranged to form the next equation. Now, realize I require you to write out every equation before
you use that equation. (mr.p?)
Yes Billy? But, what if
the equation is already written somewhere on the page? Like, at the top of the page?
That is a question I often hear near the beginning of the school year. Honestly,
there are going to be many, many equations in this class. You absolutely need to get
used to showing exactly which equation you are about to use before you use it.
(mr.p?) Yes, Ms. Langenbrunner?
What if the student writes down the equation right after they use it?
Good question. And, I am guessing you actually want to answer it. So please, go ahead.
Thank you mr.p. I do want to answer my own question. Writing down the equation that you
used after you used it skirts the whole point of writing the equation down in the first place.
The reality is that writing down your knowns and equations before you use them is a part
of the physics thinking process. When you see a difficult problem, sometimes you won’t know what
the solution looks like yet. So, if you write down your known variables and your relevant equations
at the start, that will help you process the information that you already know. That will help
you streamline your thinking and that with help you come up with the solution. We, as teachers,
want to help you build up your problem-solving skills for longer and more difficult problems
in your future. Yeah!
How’s that mr.p? Perfect. Thanks for your help.
You are welcome. Alright. A few more items to point out.
You need to substitute numbers into your equations, even though you have already written
those numbers down on the page. This decreases mistakes and illustrates what you actually did.
Every ending number in your solution must have units. Eventually we will get to some
numbers which do not have units; however, I will make that very clear when it occurs.
And, I do not require units on all numbers in the middle of equations. You can see I
did it in my solution, however, as solutions get longer and longer it really gets too cumbersome.
So, what I mean by “every ending number” must have units, is that I am okay with not putting
units in the middle of equations, however, at the end of the equation, when you get a number,
you absolutely need to put units on it. Next, you need to show all of your conversions. Soooooo many
mistakes are made because students incorrectly convert a number. Showing the conversion really
does mitigate those simple mistakes. And, always write out the unrounded answer before the rounded
one. You are less likely to make mistakes when you do this, and you might need the unrounded
number later. And yes, the unrounded density for the steel sphere is needed later in this lab.
Okay, Billy, let’s look at a line from your solution to this lab. Billy, tell me what you see.
Absolutely! Radius equals diameter divided by 2. I plugged in the diameter and solved
for the radius. And what did I do there? The radius equals 12.65 but why did I cube it? …
(I think you solved for the radius. Then you realized you were looking for the volume. So,
you used the equation for volume, four-thirds pi times radius cubed, without writing it down
and without starting a new equation.) Oh yeah, that must be what I did. So,
that is volume. Right, yeah. Because 8,479.3 cubic millimeters is the volume. But that
does not equal the radius. So, this equation does not actually make any sense, does it?
(No. Billy, how would you fix it?) Well, the radius equals 12.65. So, at that
point I would add millimeters to that number, and then I would add the equation for volume,
and plug in numbers to that equation. Yeah, that fixes it.
Thank you Billy. So, to answer the question “Why do I require you to ‘Show All Your Work!’?”
it is to help you learn more efficiently by making fewer mistakes. That is really the most important
piece. As you get further and further into your physics learning, the situations you will
be analyzing will become more and more complex, and all of the topics will build on one another.
This means your solutions will get longer and longer and longer. If you learn how to “Show All
Your Work!” at the start of your physics learning, you, and others, will be able to follow your work,
you will learn more efficiently, and you will be happier physics students. And that is what
I want for you most, to be happy physics students. So, please, please, please, “Show All Your Work!”
We are happy physics students mr.p! I am.
Actually, I am too. I know. I thank you for that.
(mr.p?) Yes, Mr. Hansen?
I would have done your first example solution differently.
(Okay. You mean where I solved for the density of the steel sphere?)
Yes. I think you should not have plugged in numbers right away when solving for
the volume of the sphere and instead take the equation for the volume of the
sphere and substitute it into the equation for density. Likewise with this equation. That way
you would get an equation for density in terms of variables only. The variables we measured. When
you hold off on plugging in numbers until the very end, you don’t have to worry about rounding errors
in the middle of the problem, you can use the equation again in multiple trials, you can use it
as a check on your work, all sorts of other (Uh Mr. Hansen!)
Yes, mr.p? I completely agree
with you. What you are describing is what I like to call “Letting go of your numbers dependency”,
however, in my experience, it is too soon in the student’s physics learning to begin talking about
their numbers dependency. Let’s hold off on that for now and have you come back to talk about that
in a future lesson. Oh boy. Uh. Woah.
Okay. Sure. I can come back later.
See you in a bit mr.p. Yes. See you in a bit.
And, that’s it for me. Thank you very much for learning with me today, I enjoy learning with you.