George Lucas Educational Foundation
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What do you do when, after dividing two whole numbers all the way to the ones place, you're still left with a number? The lazy solution for mostly everyone I know seems to be to put an "r" next to the quotient, put the leftover number, and call it a day. Well, stop it. This instant. Like, now.

As a math teacher, I often find myself trying to elevate my students' knowledge to the point where the transition into the next level isn't as difficult. Rules don't necessarily change, but they take a new form, and the less friction we can create as they become more mathematically proficient, the better.

For instance, students in the first grade might think there are no numbers left before the number zero -- until they get to the sixth grade where they have to learn about negative numbers. They might learn in the eighth grade that they won't "normally" see a graph that's a sideways parabola, which is true because that's not a function. However, as they start getting into calculus, they'll see graphs that break the mold of what a graph should look like time and again, and they'll have to interact with those graphs, too.

3 Ways of Looking at a Remainder

But with remainders, it's a little different. We can much more easily create a consistent set of understandings by simply eliminating the idea of "remainder x" and replacing it with any one of the three methods below. In fact, I encourage at least two of these.

1. Remainder as a Fraction

This is the easiest out of the three because, once you've gotten through the whole-number portion of the quotient, you simply ask your students, "What's left?" They'll respond with the assumed remainder. What you'll ask next is, "What have we been dividing with?" They'll hopefully point out the divisor. Once you've made the connection between what the remainder symbolizes and the divisor, they have another way of looking at fractions. Now, they're actually looking at fractions as the representation of what happens when numbers don't fit neatly into groupings we've chosen. From there, this next one would also make sense.

2. Remainder as a Decimal

A few students ought to get curious as to what happens if we want to represent this non-whole number in our base or, simply put, as a "decimal." If the students keep dividing, they'll eventually get a lesson in number sense and judgment. For instance, if a number passes the hundredths place and the situation calls for a money-friendly number, then perhaps rounding at the hundredths makes sense. If the number continues with no discernible pattern, that's where teachers and/or students can have a discussion about why math isn't always so neat and organized as they thought it was. However, if there are only one or two decimal places, it's a much more elegant solution -- and more readily useful in practically every case!

3. Remainder in Modulo Operations

If you'd really like to have some fun (and introduce a little coding to students), you can introduce your students to the idea of "mod." For those unfamiliar with this math term, "mod" means that any two numbers leaving the same remainder when divided by a common divisor are congruent. For example:

100 = 88 (mod 6) (which we read as “100 is congruent to 88 mod 6")

Why? Because, when divided by 6, both 100 and 88 leave a remainder of 4. This will help when students are asked to change from base 10 (decimal) numbers to base 16 (hexadecimal) numbers. In other words, it pushes the limits of what students actually know about our number system, and it's easily transferrable from the younger to the elder grades.

Unlike "remainder x."

Apples in Baskets

I promise I'm not begrudging anyone who uses the old style of remainders. I'm sure there's some usefulness for someone, especially when you're just beginning long division. Yet a part of me feels that, even if we told our sixth grade students that they should use a more precise rational expression, they'd be better off. For instance, if we wanted to put 13 apples into 2 baskets and asked students to represent that, they can say it as either:

  1. Each basket gets 6 apples, and we can slice the remaining apple in half so that it can fit into each basket.
  2. Each basket gets 6 apples, and if we have one more, we put something in a third basket.
  3. We can fit 6 apples into each basket, and we have 1 apple left.

In my experience, the "remainder 1" in this problem always keeps us stuck in option #3, which often leads nowhere. Let's put the "remainder" to the side and, if we don't know how to handle it, round it up to the nearest whole number. Our future mathematicians will thank you later.

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Mike Petty's picture

As a former high school math teacher who now K - 12 in technology, I can appreciate this post. This is one of many small changes in the introduction of concepts that will make future learning much easier for students.

I would often hear from students who struggled with math that what I was teaching contradicted something they learned years earlier. Negative numbers, as you mention, was one example. The definition of basic shapes was another.

Many times it wasn't presented incorrectly by the teacher. It was just the fact that a student could get by with a limited or incorrect understanding of what was taught in elementary. The misunderstanding didn't matter then, but the learning at that early age was very difficult to correct later when it did matter.

CCSSI Mathematics's picture

If in middle school you are still teaching remainders and/or students are still expressing answers with remainders, that's certainly a problem, but remainders are an appropriate stepping stone to higher arithmetic. If you insist that remainders are simply an anathema, then division won't be taught starting in Grade 3, where it belongs.

An analogy is that students don't have to know p is irrational and its decimal representation is non-terminating and non-repeating to understand its meaning and to start using it as early as Grade 5.

As for bases and modular arithmetic, they are strangely absent from Common Core. That's why most American students won't be seeing standard problems like these, which put students' understanding of remainders to work:

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