Here's a problem that came home in my daughter's backpack:

*Ebenezer has 97 lunch trays. He will make four stacks of lunch trays. He will put the same number of trays in each stack. He will put as many trays as possible in each stack.* *If Ebenezer does this, how many trays will be in each stack? How many trays will be left over?*

Let's consider the first question first.

*How many trays will be in each stack?*
I found the answer 24 by starting down the road of calculating 97 ÷ 4 far enough to see that result would be "24 and some stuff." (Too late, I remembered that I actually knew 4 × 24 = 96 offhand—it was still a good check on my answer.)

Thus my approach was to pursue, just far enough, a strategy of "divide and round down." If you work with computers, then you probably know that the operation "divide and round down" equals a computer command known as "DIV." For example, a computer will tell you that 16 DIV 3 = 5, and one way to check this answer would be to calculate 16 ÷ 3 = 5⅓ and round the result down.

I doubt this is how a computer actually calculates the value of 16 DIV 3, but "divide and round down" gives the right answer and shows the relationship between DIV and ÷, which is what I want to write about today.

A few more examples just to clarify how DIV works:

32 DIV 7 = 4

54 DIV 6 = 9

3 DIV 4 = 0.

A friendly term for the DIV operation might be "arraying." Ebenezer is trying to

*array* 97 things in a 4-by-something array. It's to be understood that when you are arraying things, the equal groups are to be as large as possible, and there might be some articles left over. I think the word "arraying" has a discrete connotation that helps convey its meaning. We array pieces of silverware when we set the table, but we don't "array" the milk when we pour some into each child's cup. (Update 9/4/17: Maybe you could think of DIV as a "packaging up" operation. That phrase avoids the spatial connotations of arraying. For example, if an elevator fits 6 people and there are 20 people who need to get to the roof, then it is a packaging up operation more than an arraying operation. Anyway.)

One nice thing DIV enables us to do is to write down a formula for remainders. For example, if we know that 97 DIV 4 = 24, then we can multiply the 24 by 4 (result: 96) and then what's left over is the remainder; in symbols,

remainder = 97 − [4 × (97 DIV 4)].

More generally, we have the following formula for remainders in a whole-number problem:

remainder = *m* − [*n *× (*m* DIV *n*)].

There is a second well known computer operation, called MOD, which finds the remainder; so we could write

remainder =

*m* MOD

*n*

or using the previous expression,

*m* MOD

*n* =

*m* − [*n *× (*m* DIV *n*)].

Rearranging this, we can say for any integer *m* and any nonzero integer *n*,

*m* = *n *× (*m* DIV *n*) + *m* MOD *n*.

This is the whole-number analogue of the much simpler relationship for rational numbers *r* and *s* ≠ 0,

*r* = *s* × (*r* ÷ *s*).

Notice there's no remainder here; "division with remainder" is a contradiction in terms.

Of course, the relationship

*r* =

*s* × (

*r* ÷

*s*) is simpler only if you understand rational numbers (or at least fractions). The curriculum in late elementary school often doesn't do enough in that respect, leaving students in middle school stuck with cumbersome (or even faulty) whole-number ideas about operations.

Naturally there are many word problems in which arraying (rather than division) is called for—because a cup of flour can be partitioned any which way, but lunch trays, marbles, wheelchairs, and teachers can't be partitioned and remain lunch trays, marbles, wheelchairs, or teachers. So in a word problem, the context determines the best answer to give—whether this be the exact quotient of two given numbers, or the whole-number part of the quotient, the quotient rounded to the next greatest whole number, or even the remainder. Getting the right answer to such problems is better thought of as a modeling skill than as a calculation skill.

In fact, I suspect that the existing curriculum spends too much time having kids practice bare whole-number division problems with remainders: that confuses the subject of arithmetic by giving division two incompatible meanings. Of course, I would give students plenty of distributed/repetitive practice with multi-digit long division; I'd just set things up so that in all those practice problems, either the divisor divides the dividend evenly, or else the answer is to be expressed correctly as a fraction or decimal. (So that, in both cases, the true relationship

*r* =

*s* × (

*r* ÷

*s*) holds. You should always be able to check a quotient by simply multiplying.) Meanwhile, situations with remainders would be prevalent in the word problems students work on.

Perhaps that approach is simplistic, but certainly I'll say that I was appalled once to see my daughter working on a giant drill worksheet with dozens problems like this:

5 ÷ 4 = 1 R 1.

My daughter was terribly confused at first. "Isn't 5 divided by 4 equal to five-fourths?" she asked. I answered her by saying, "Yes it is. You are right. But for tonight, just write what they want." I didn't know what else to say. That worksheet penalizes students who have correctly learned basic fraction concepts and division concepts—concepts which, unlike whole-number concepts, are essential for the transition to algebra.

What a terrible and unnecessary conflict between ideas. There is a need for practicing calculations like 8,995 ÷ 7 = 1,285 so by all means, provide distributed/repetitive practice with those kinds of problems. But it can't be educational for students to rehearse statements like "5 ÷ 4 = 1 R 1," which lie about the division symbol in ways that grade-level students can detect.

I'll close with some better news, specifically in reference to

this related earlier post about division. On a recent Saturday School I was elated to see, in

a series of workbooks written in 2001 by my colleague Marsha Stanton, this page:

This is the best case I've seen so far of using a true equation to represent objects arrayed in equal groups with some objects left over. This would be worth doing as a way to represent word problems, as I wrote in the

earlier post (one-page PDF at

tinyurl.com/notdivision).