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I'd like to introduce you to one of the magical mathematical skills that we teach students at EAB starting in the Early Years Program: Number Bonds!  A number bond is a simple picture of a part-part-whole relationship that looks like this: 

What makes it powerful is that the same 7 can break apart in many ways: 3 and 4, 2 and 5, as well as 1 and 6.  You can see all of these represented below.  

Instead of memorizing seven as a fixed fact, the child comes to see seven as flexible; something that can be taken apart and put back together however a problem requires. That flexibility is the whole point, and it is the foundation of what teachers call number sense.

The thing that's important in later grades is that a single number bond doesn't just give one fact; it gives a whole family of them. From the bond that joins 3, 4, and 7, a child can read off four true sentences: 3 + 4 = 7, 4 + 3 = 7, 7 − 3 = 4, and 7 − 4 = 3. Once children see that these four sentences are really the same relationship viewed from different angles, something important clicks: addition and subtraction are not two separate skills to be mastered one at a time. They are different ways of describing one picture. The National Council of Teachers of Mathematics has long emphasized exactly this kind of connected understanding over isolated fact memorization.

I was in a third-grade classroom last week, and students were working on this problem:

____ − 389 = 74

That's a genuinely hard problem to attack head-on. What number, with 389 taken away from it, leaves 74? A student trying to "just subtract" is stuck, because there is nothing obvious to subtract. Some of our students reason it out on a number line, which works beautifully. But look what happens for a child who grew up drawing number bonds. 

They recognize the structure instantly: 74 and 389 are the two parts, and the blank is the whole. And from any number bond, you can always write the family of sentences — so the blank simply equals the two parts added together:

____ = 389 + 74 = 463

The hard subtraction problem has quietly become an easy addition problem. That is not a trick or a shortcut the child memorized. It is the same part-part-whole idea they've been drawing since they were five, now doing real work on a problem three grades up.

And it keeps paying off. Those same fact families reappear with multiplication and division: the bond joining 3, 4, and 12 gives 3 × 4 = 12 and 12 ÷ 3 = 4, which is exactly what a student needs to solve a missing-factor problem like 6 • ___ = 42.

In pre-algebra, this type of reasoning becomes the backbone of equation-solving. We replace the blank spaces in equations and question marks in number bonds with variables, showing students that they already have the tools they need to solve algebraic equations. A student who has internalized number bonds sees an equation like x − 389 = 74 for what it is: x is the whole, and the whole equals the sum of its parts. Many of us were taught to "move the number to the other side and flip the sign" — a rule that works but that few of us could actually explain. There is no more mystery or disjointed rule; there is just a relationship to read. 

The same instinct powers quick mental math, too.  When we add 8 + 5 by splitting the 5 into 2 and 3 to "make a ten" (8 + 2 = 10, then 10 + 3 = 13), we are using a number bond.

So the little circles are not a detour on the way to "real" math. They are where a great deal of later math is quietly rooted. When our youngest students decompose and recompose numbers, they are building the flexible, connected understanding that turns hard problems into approachable ones for years to come.