Confusing variables with units in simple equations

Question

I have noticed that beginning algebra (or pre-algebra) students can easily make the following type of error, confusing variables with units when being introduced to equations for proportional relationships.

Say they are presented with an equation $d=6t$ and told that it represents how fast a person can run, with $d$ representing distance in miles and $t$ representing time in hours. The student then reasons as follows:

$d$ is miles and $t$ is hours, so the equation says that 1 mile is the same as 6 hours, or in other words the person runs 1 mile for every six hours.

They might make the same type of error in the other direction: if told that someone runs 10 miles for every two hours they might reason

$d$ is miles and $t$ is hours, so the person runs $10d$ for every $2t$ and the equation is $10d=2t$.

What are some ways of helping them correct this mistake?

Answer 1

A practical suggestion: make sure your own equations are dimensionally consistent, and your students will be more likely to make theirs consistent as well, thus avoiding mistakes like these.

In particular, your equation $d = 6t$ is not dimensionally consistent. If I plug in, say, $d = 1\text{ mile}$ and $t = 1\text{ hour}$, I get the obviously nonsensical equation $1\text{ mile} = 6\text{ hours}$. Which is what the student in your example got too, and then tried to somehow make sense of. Arguably their mistake was in trying to make sense of nonsense, but it's an understandable mistake if they haven't been taught any better.

OK, but enough polemic. If $d = 6t$ is nonsense, then how should you correctly express the idea that a person runs 6 miles in an hour? Simply like this:

$$d = 6\frac{\text{miles}}{\text{hour}}\, t$$

Now you, or your students, can in fact plug in, say, $t = 1\text{ hour}$ and solve for $d$:

$$\require{cancel} d = 6\frac{\text{miles}}{\cancel{\text{hour}}} \cdot 1\,\cancel{\text{hour}} = 6\text{ miles}$$

Better yet, they can check that the result they got in fact fact has units that make sense. If they had instead obtained a result of, say, $d = 6\text{ hours}$, they would've immediately known that they'd made a mistake somewhere, since $d$ is supposed to be a distance, and an hour is not a unit of distance!

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I do understand that my suggestion above might not seem practical at a glance, since it basically amounts to completely changing the way you teach basic algebra whenever physical quantities with units are involved. You might even have a textbook that teaches things the same way you currently do, and you'd have to either find a new textbook or tell your students "don't do it the way the book says, do it the way I say" (which, while sometimes a possible solution if you must work with a subpar textbook, is far from pedagogically ideal). Your students may also have previously learned to do things differently, and you might need to get them to unlearn bad habits like leaving units out of equations. Heck, you might have to force yourself to unlearn these habits first, too.

But I would still suggest at least considering this approach. Maybe try it out on one class with some particularly struggling students as an experiment. (Do tell your students that you're doing an experiment, especially if you've previously taught the same class to do things differently, or they'll just be confused!) I am quite confident that, despite whatever initial difficulties you might have, in the long run this will get you better results and help your students better understand what's happening with equations like this.

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Ps. Also make sure that your word problems actually have answers that make sense, so that your students can use their common sense to check their answers. And teach them to refine that common sense using techniques such as dimensional analysis (as described above), order-of-magnitude estimation and other approximation and consistency checking methods. This is a valuable skill that will serve your students well in their life, and one that mathematics education all too often neglects (or sometimes even actively harms!).

Answer 2

There are two things that can be done to really drive this point to someone first learning algebraic equations.

1. Focus on how the units of both sides of the equation must be equal. This might seem elementary, but you've noticed your students reasoning that somehow time can be equal to distance which doesn't really make sense. It's just not obvious to someone first learning algebra. Try to make this a fundamental concept that they consider when they approach any problem. This is incredibly useful for learning how to approach more complex mathematical expressions.

2. Make the units of the problem very explicit in the statement. This supports #1 so the student can use that information to check their solution.

Here's an example problem:

Alice is late for school and is running to make it to class before her math exam begins. An expression for how far Alice has traveled is given by the equation d = 6t where d is the distance in miles, t is the time in hours. What units must the number 6 in the equation have to satisfy the equality? How fast is Alice running? (answer: miles/hour, 6 miles/hour).

Here's how I would approach the other example from your second problem.

if told that someone runs 10 miles for every two hours they might reason "d is miles and t is hours, so the person runs 10d for every 2t and the equation is $$10d=2t$$.

Let's say they try to make this assertion. Now the student should check their units. If the number 10 has units of miles and the number 2 has units of hours as posed by the problem, the units of the expression they created are $$miles^2 = hours^2$$

This doesn't match up, so we must have gone wrong somewhere. Then I would approach the problem by trying to determine where the constants and the variables need to be in the expression to create equal units on both sides.

Answer 3

I think the way to help this kind of student is to get them in the habit of reading the equation correctly. The equation $d=6t$ does not say that

1 mile is the same as 6 hours,

it says that

the number of miles is the same as 6 times the number of hours.

In other words, if you want to calculate the number of miles, take the number of hours and multiply by 6.

So in one hour, the person runs 6 miles.

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In the other direction: if someone runs 10 miles for every 2 hours, then they run 5 miles each hour, so

the number of miles is the same as 5 times the number of hours,

and the equation is $d=5t.$

Answer 4

One possible method is to have students simply do t vs d value tables. "If t is 1, what value does d have to have to make the equation true?" So starting with given values for t, figuring out d is just a multiplication by 6.

Assuming they are comfortable with that, then start with a value for d, and ask what value of t will make the equation true. "If d is 12, what value of t will make the equation true?" "What number times 6 equals 12?" Obviously these would then all be division problems, and hopefully the students remember their 2nd and 3rd grade "fact families"!

This would, I hope, avoid worries I have about whether the students understand concepts like conversion factors and unit rates.

Answer 5

Personally, I would emphasize translating natural statements very carefully, one word at a time. Starting with the fact that the verb "to be" (is) is the thing that translates to the equals sign in an assertive statement. So:

someone runs 10 miles for every two hours

The first thing I'd note is that there's no "is" verb in this statement. It's somewhat vaguely/sloppily expressed. We can improve on it like so:

someone's running speed is 10 miles for every two hours

Okay, now we have an assertive statement with "is". Ideally we'd carefully define a variable for the quantity in question: $s$ = speed, in miles per hour. Importantly we note that the "is" translates to equals, and "for every" (or "per") translates to division. Hence:

$$s = 10 \text{ miles} / 2 \text{ hours} = 5 \text{ mph}$$

Most elementary textbooks have blocks of exercises on this translation skill, so you can lean on that resource for practice. Generally the statements there are more clear then in the OP example, and I'm likewise careful to phrase my examples precisely for beginning students.

If the instructor's intent is to present more vaguely-formed statements and let the students grapple with that, then it's by nature more challenging. I will also say that I've found proportional problems in a basic arithmetic class to be among the most troublesome, precisely because no book or exercise ever explains that things are proportional -- students are expected to just intuit the proportionality in each separate case.

Answer 6

I posted an answer earlier, but after more thought, here's a different approach that I think might be better.

Student: $d=6t$ means 1 mile is the same as 6 hours.

Teacher: Really? So if you plug in 1 mile for $d$ and 6 hours for $t$ the equation comes out true?

Student: $1=6\cdot 6 =36$... Wait that's not right.

Teacher: Plug in 1 hour for $t.$ What comes out for $d$?

Student: $d=6 \cdot 1 = 6.$ So 1 hour is 6 miles.

Teacher: Exactly. It's the opposite of what you said initially.

Answer 7

I'm coming from a physics background. I avoid writing numbers for any measured quantities into equations until the last step where I plug in values to get a final answer. This is because it can be difficult to remember which number have quantities, which can make finding errors harder. In the equation $$d=6t,$$ it is not immediately obvious that the 6 has units of miles per hour. Bare numbers usually don't have units unless explicitly written out. The left side looks like it has units of distance, while the right side looks like it has units of time, which doesn't work [1]. Even if I knew that the speed of the running child was 6 miles per hour, I would still write $$d=v\times t.$$ The natural way a student would read this is "distance is velocity times time," which would be correct. All measured quantities (especially those with units) are given symbols to distinguish them from bare numbers that don't. Another advantage to using only symbols for quantities is that it allows for error checking during algebraic manipulation. If I'm doing a long derviation and I find that I've written $x + t$, where $x$ is position and $t$ is time, I know I've made a mistake earlier and can go looking for it. If I write $5 + t$, it's much less obvious.

Another example is the acceleration equation: $$d = \frac{1}{2}at^2.$$ The $1/2$ is a mathematical constant with no units, so it needs no symbol. A quick glance shows that both sides have units of distance, which makes subsequent algebraic manipulation easier to reason about.

[1] I'm ignoring the high-energy physicists who set $c=1$ in their equations.

Answer 8

I would say that your students are completely right:

In general life it's completely normal to write six seconds as $6s$ or thirteen meter as $13m$, so thinking that $6t$ is some way of writing "six (t)s" where $t$ is some time or space unit, that's just common sense.

In order for you to avoid this, you might write the dot operator as the multiplication, so not $6s$ but $6 \cdot s$. It's a very small burden for you, but a lot clearer for your students.
Once you start dealing with more complex equations, like $y=mx+q$ or $ax^2+bx+c$, you might start dropping the dot (but in case of doubt, you might always choose to put it back).