## Add and Subtract Odds and Evens

Textbooks are often pretty good about how they introduce even and odd numbers.

Presentations generally consistently involve a description of an even number of objects as a collection that, when separated into groups of 2, has no objects left over. An odd number of objects is a collection that, when separated into groups of 2, has exactly one object left over. Students are also generally given visual and/or tactile reinforcement of these concepts.

What's nice about this approach is that it provides for a clear understanding of even and odd numbers that does not conflict too much with later learning and gives students a foundation on which to build more sophisticated thoughts about the concepts of even and odd.

Textbooks, however, often fail to follow through on this great start by using inductive methods or, even worse, simple rules to present the topic of adding and subtracting even and odd numbers. Students are often expected to just memorize the rules (even + even = even, etc.) or try a few test cases (2 + 4 = 6, etc.) and proclaim after three or four examples that the rules are, indeed, valid for all numbers to infinity.

But consider the following approach, which builds on this clear understanding of even and odd numbers and gives students the opportunity to see real mathematical thinking.

Suppose I have two boxes on my desk. In the first box, all the objects have been separated into groups of 2. I have one left over, which I have placed on top of the box.

Although this box has a certain number of items, we can imagine that we do not know how many objects I have in the box--only that the collection has been separated into groups of 2 with one left over. So this box represents an odd number.

In another box, objects have also been separated into groups of 2. There are none left over here, however--there are no objects on top of the box. So this box represents an even number.

I can imagine that I do not know how many objects are in either box. I only know that one box represents an odd number, and the other box an even number. The only critical characteristic about any of the boxes is whether or not it shows 1 left over (odd) or zero left over (even).

If I combine the objects in the two boxes (i.e., add the values), I do not change how the objects are grouped--all the objects in the two collections are still separated into groups of 2. I need only add the leftover amounts to see if I will have one left over or none left over.

Since one left over + none left over = one left over, my sum is an odd number. And since we could find this result without knowing the actual amounts involved, we can say that it works for any amounts. We have found a rule that always works: odd + even = odd.

The other rules are easy. Keep in mind that when you add two odd numbers (1 left over + 1 left over), you are forming another group of 2 with none left over. Here are the rules:

Even + Even = Even (0 + 0 = 0)

Even + Odd = Odd (0 + 1 = 1)

Odd + Even = Odd (1 + 0 = 1)

Odd + Odd = Even (1 + 1 = another group of 2 with none left over)

And because subtraction is simply the opposite of addition, the rules for subtraction are the same. The equations above can be reversed to show that this is true.