Sometimes when you are teaching, you have to simplify a problem so that students can understand the concepts involved. I will explain. When I was a teacher, I was trying to explain to my Honors Class (I think chemistry, might have been physics. It’s been awhile) how to do a unit conversion using a method called dimensional analysis. It’s a pretty common way of solving equations that is used in the physical sciences.

I first learned the dimensional analysis when I was in the Navy at NucField A school and at Nuclear Power School. It’s handy for solving a lot of things. Dimensional analysis is a method for solving various problems that, once mastered, allows for the rapid solution to unit conversions, various physical problems (like Ohm’s law), medication dosage calculations, and more. It reduces calculation errors and is a very handy skill to have. I use this method all the time to calculate drug dosages, and used it as a firefighter to calculate hose pressures and other useful numbers.

I was teaching it to my students by giving the students a list of things I wanted converted from one item to another. The worksheet that I gave them was a list of problems that were easy to solve, but included the following instruction:

Show all of your work, including the proper setup of the dimensional analysis method. Your work is part of your answer, and any problem that does not include the showing of your work in the proper format will be marked as incorrect and will receive no credit.

The questions were things like:

- Convert a $5 bill to nickels
- How many toes would 22 people have?
- How many legs would 123 ants have? (they each have 8 legs)
- etc

So one of my students answered:

- 100
- 220
- 984

And promptly got a zero for a grade. Yes, the math was correct, but I wasn’t looking for the mathematical solution, but a solution that showed me that he had mastered the method of dimensional analysis. Anyone with fourth grade understanding of arithmetic can tell you that 22 people have 220 toes. I knew that they could do simple math, because it was a requirement to have already passed Algebra and Geometry as a prerequisite to even be in my class. This was an honors course where students could receive college credit at the end of the course.

It was important that they understood the concept so when we went on to more complicated problems, they would have the skills needed to solve them. It wasn’t about the math of that particular problem, it was about knowing HOW to use dimensional analysis. That way, when you get a problem that goes like:

A sample of calcium nitrate, Ca(NO

_{3})^{2}, with a formula weight of 164 g/mol, has 5.00 x 10^{27}atoms of oxygen. How many kilograms of Ca(NO_{3})^{2}are present?

The problem can be solved without too much difficulty. The easy problems were not a test of math ability, they were a means of learning a new method for applying math skills that the student already has. A “learn to walk before you try to run” sort of thing.

The child’s parent wrote me and demanded that he receive full credit because he got the correct answers. I tried pointing out the instructions and explaining the reasons behind showing your work. No go. The parent argued that “in the real world” no one cares how you got the answers, just that you ended up with the correct answers. I tried pointing out that, this being school, demonstrating that you have mastered the method is more important than getting the correct answer. The parent continued to argue and demanded that the student receive credit. I refused. They even told me that they would get a lawyer involved. I told them “good luck finding anyone that will support you not showing your work on math homework, when the instructions clearly required it.” Then I told them if lawyers were to be involved, I would be happy to give them my attorney’s number, and their attorney could call mine to arrange a meeting. They hung up on me.

So the parents went to the principal. Nope. They went all the way to the school board, to no avail. The parent finally pulled the kid out of my class and put them into a low level environmental science class.

In this case, the parent did the child no favors.

EDIT: I am editing this to give an example of how dimensional analysis works. Here is the example:

Convert 1 week into seconds:

*Terms on the top AND bottom of the equation cancel out, leaving: (1*7*24*60*60s)/1=604,800 s

## 21 Comments

## Olguy · May 25, 2023 at 5:04 am

So, the Parent pulled the kid because you were attempting to educate them?

Well! β¦ there is the Hill to Die on.

That same parent would bitch about drag queens in school and LEAVE there kid there in that school.

β¦β¦some kids parents.

## Divemedic · May 25, 2023 at 7:22 am

I suspect that the parent, who claimed to be an engineer for Florida Power and Light, was the one who actually did the child’s homework and was angry that they got all of the questions wrong.

## wojtek · May 25, 2023 at 8:56 am

So everything depends a little on whether the sample questions you listed above are oversimplified for the sake of presenting them on the blog or if they really represent the questions you asked in class. But if it is the latter then in my opinion you are wrong.

First of all you are incorrect that this example illustrates the issue we were talking yesterday about, that is the need to oversimplify the answers for students in some cases. The example you gave is in fact the opposite of that: you have (in my opinion unnecessarily) overcomplicated simple questions about multiplication and division as “dimensional analysis”, which led to problems that could have easily been avoided. This is a reflection of the other end of the spectrum of educational issues. Yesterday, we were looking at teachers who simply did not have the subject matter expertise and were only trained in “education”. Now, let’s be clear: I discovered your blog a few days ago and I know nothing about your background – but from my own experience I would say that your example is symptomatic of the teachers who have the needed subject expertise, but do not have the necessary training in didactics, pedagogy, or whatever you call it. Personally I think teachers need both.

Second, I think that if you really gave the student zero credit it also was wrong. If the student answered all questions correctly and there were more than 2 or 3 of them, then you should have assumed that it was not by chance but rather that the student (or whomever was solving these) understood the pattern behind your questions. Here your instruction that not showing work will count for zero credit is what in Poland we call dupochron. It serves no pedagogical value. The purpose of teaching is to provide students with understanding of the subject we are teaching. Not to make them follow instructions verbatim.

## Divemedic · May 25, 2023 at 9:05 am

Except you are misunderstanding the point of the lesson. Was the point of the lesson to teach them how to multiply 10 toes times 22 people and arrive at 220 toes? Or was it to teach the student how to apply dimensional analysis (using simple math problems) in order to arrive at the correct answer?

The expectations of the assignment were clearly laid out. I will repeat them here:

Show all of your work, including the proper setup of the dimensional analysis method. Your work is part of your answer, and any problem that does not include the showing of your work in the proper format will be marked as incorrect and will receive no credit.Note that the correct answer was not the numerical one. According to the instructions, it was the setup and use of the dimensional analysis that was the correct answer.

## Don Curton · May 25, 2023 at 12:05 pm

In college, for chemical engineering, we had various tests where the “correct” answer was to put down the full equation that was applicable to the problem, with all values entered for each variable in the right place, with the right units. The answer was not the end result of the calculation, but the equation (or formula) correctly written.

Had I simply answered with a numerical value (correct or not), I would have received a zero. Same concept, different skill level. Completely agree that the student most likely didn’t do the work and deserved the zero.

That said, plenty of times my kids brought home assignments that made zero sense and used some form of math that was not correct, not logical, not right. Had discussions with teachers who went into the whole “this is part of a new learning process” BS while teaching incorrect math.

As an engineer with over 30 years in industry, I still show all my work, including every step in the process and every unit conversion. Old habits die hard.

## wojtek · May 25, 2023 at 1:14 pm

I think I understand very well: this is what I called dupochron. You are legally covered. But my personal opinion is that – by trying to simplify DA – you have effectively overcomplicated what is a simple multiplication/division problem and tried to sell it as dimensional analysis. This is a mistake because when your audience is sophomores and juniors, asking for “showing your work” for */ is really pointless. Heck, I think that asking for showing your work for */ problems in 2nd grade of elementary school is silly. This is because an assumption that everything for everyone needs to be explained in exactly the same terms is just false. With that attitude, if you were a math teacher Ramanujan would fail your class :)))

## Divemedic · May 25, 2023 at 1:51 pm

OK, here is how the toes problem would look:

Understanding how to do this is the basis for more complicated problems. A great example of this is in drug dosage calculations. When I was in nursing school, we were given a math quiz once per week. We were given 30 minutes to complete a 12 problem quiz. You had to get all of them right, or you failed. You got two attempts at each quiz (different problems each attempt). If you failed any month’s quiz after the second attempt, you were out of the school. Here is a couple of sample questions:

1. You patient has a solution of Heparin running. The solution is 25,000 units in 500mL of D5W. It’s infusing at a rate of 25 mL per hour. How many units per hour is the patient receiving?

2. You have 400mL of Dopamine in a 250mL bag of NS. The order is for 5 mcg of Dopamine per kg of body mass per minute. The patient weighs 132 pounds. How many milliliters per hour will you infuse into the patient?

3 The doctor orders you to give 20 mg twice per day. You’re supplied with a bottle that is labeled 5mg in 50ml. How many tablespoons of medication will the patient receive per day?

Remember, there are 12 of these, you have 30 minutes to do them, and you can only miss one.

## wojtek · May 25, 2023 at 2:45 pm

I have an even better solution: 22 x 10 = 220 π

It’s a single multiplication, because nothing in the problem prevents you from doing “DA” with the assumption of 10 toes per person, instead of 5 per foot. And then it becomes multiplication and is a correct solution. Same with ants and legs. $$ and nickles almost the same b/c of an unfortunate choice of $5 bill and 5c coins and commutativity of multiplication. Each one of them boils down to multiplication/division. There is no element of regrouping or a multi-step solution needed.

And to add to your other point, whenever possible this multiplication/division is the fastest way to solve such problems, which also means most practical.

So, to close this discussion: it is all about the examples and how they increasingly introduce students to complexities involved in the new technique. It’s OK for the first problem or two to be */, provided the next problems add new complexities. Only then students can realize that DA is not the same as multiplication. I am assuming that your HW problems progressed along the edit problem you showed.

## Divemedic · May 25, 2023 at 2:53 pm

You are assuming that the point of the problems was to test your ability to perform multiplication. It is not. As I said, it is assumed that you know multiplication. Why would I give a student an assignment to do something that I know they can do? That would be pointless.

The assignment was an attempt to teach a new method, DA by using an easy problem that we can use the new method to solve. The concept is called scaffolding.

Let me try one more time to explain to you. When you learned multiplication, they made you learn the multiplication tables at some point, even though it is easier to think of multiplication as repeated addition. Why memorize 4 times four equals 16, when it is easier to learn 4+4+4+4=16?

The answer is because learning the tables makes 123*254 a more manageable problem than trying to add 123 to itself 254 times.

## Echo Hotel · May 25, 2023 at 3:08 pm

Good effort, DM.

“Some men you just can’t reach…”

## wojtek · May 26, 2023 at 10:12 am

“You are assuming that …”

We will not be able to finish this discussion if you will be making my assumptions for me π No, I am not assuming, I am claiming that by giving them too simple problems, effectively multiplication problems, you have conditioned them to discover the simplest pattern occurring. Like if you were teaching regression analysis and gave them entire HW consisting of 2-point problems and failed people for just giving you the equation of the line that passes through them. B/c you want to see the method. No. You can do this for 1 or 2 problems, but then you need to escalate the difficulty. If you don’t, then your students will assume (mind you, not me, your students) that this is really about multiplication. Not regression analysis. Or DA. Not productive.

But more importantly – it’s not a good example of why anyone would want to oversimplify. In fact I think the situation you described actually proves that in this case it really did not work. And that was my main point.

## Divemedic · May 26, 2023 at 10:17 am

1 That was why they were required to show work.

2 There was a lengthy lesson that preceded this assignment.

3 Every student but this one got it. Some people would rather get parents to fight their battles than simply learn what they are intended to.

## exile1981 · May 25, 2023 at 10:55 am

When we design piping or pressure vessels we have to show our work to prove our design to the regulator. They dont take my word, they verify my calcs.

## wojtek · May 25, 2023 at 1:15 pm

False comparison – they do not make you show your work that 5 metric tons is 5M grams π

## wojtek · May 25, 2023 at 1:19 pm

Re Edit: As I said before, everything boils down to the actual examples you asked for in class. Example from your edit is a good example for DA. Examples you gave first, about toes and legs, are not.

## Rick T · May 25, 2023 at 1:23 pm

Not doing unit conversions properly is how you crash spacecraft into Mars or sink submarines, so yes, Showing Your Work is important.

When I was doing performance model programming for an electrical power system we finally made it a group standard that everything would be done in fundamental units: meter, volt, amp, farad, henry, etc. We had too many subroutines passing data around to do it any other way.

## wojtek · May 25, 2023 at 2:48 pm

It’s not that they didn’t do the conversion properly. They simply didn’t convert π

## Vlad · May 25, 2023 at 2:29 pm

Were those Chernobyl ants with 8 legs?

Being insects, usually thereβs only 6. π

## Divemedic · May 25, 2023 at 2:33 pm

True. In my problem, they had 8 and identified as spiders.

## lynn · May 27, 2023 at 3:12 pm

I write and sell engineering software for a living. We have random numbers throughout our code because the programmer did not bother to spell out what the number consisted of. I call them magic numbers. And, most of the older numbers (our code dates back to the 1960s) are single precision with just four or five digits of precision. I shoot for 14 or 15 digits of precision now.

## Steve the Engineer · May 28, 2023 at 6:34 am

I don’t remember where I learned this if it was high school AP physics or chemistry or an “101” type class in engineering school – but when my kids were in maybe 4th or 5th grade and starting to learn the metric system and other measuring units conversion I tried to teach them this. One got it right away (they’re twins) and it took the other a while, but eventually it took hold – I’m pretty sure the one that takes a little longer to understand stuff like this will retain the knowledge whereas the quick learner will need a refresher from time to time).

Anyway while that is pretty basic stuff it is amazing to me sometimes how a lot of people can’t think in that way. “wut? convert furlong-stones to inch-pounds? huh?”