The part-part-whole model shows how numbers can be composed of smaller parts and decomposed from wholes. A whole number can be broken into two (or more) parts, and the parts can be combined to make the whole. Understanding this relationship is fundamental to seeing how addition and subtraction are connected.
Use visual diagrams showing circles or boxes divided to show parts and the whole. Use objects that can be physically separated and combined. Label parts clearly as 'part' and 'whole.'
From your work with combining sets and separating sets, you know two things: putting groups together gives you a larger group, and taking part of a group away gives you a smaller group. The part-part-whole model gives you a way to see those two operations as aspects of the same underlying relationship, rather than as separate procedures. If you know the parts, you can find the whole. If you know the whole and one part, you can find the other part. Everything connects through the relationship between the number and its components.
Think about the number 7. You might see it as 3 + 4 (three is one part, four is the other, seven is the whole). But it's also 5 + 2, or 6 + 1, or even 7 + 0. Each of these is a different way of breaking the whole into two parts. The part-part-whole model treats all of these as versions of the same structure — a whole, and its two parts — rather than as separate addition facts. When you draw a diagram with a big box (the whole) and two smaller boxes underneath it (the parts), you're making that structure visible and giving yourself a thinking tool you can use whenever you're solving a problem.
Here is why this matters: addition and subtraction are not really opposites — they are the same relationship seen from different directions. 3 + 4 = 7 and 7 − 4 = 3 are both true because 3, 4, and 7 are related through the same part-part-whole structure. If you know any two of the three numbers, you can always find the third. Addition finds the whole when you know both parts; subtraction finds a missing part when you know the whole and the other part. This is one of the most important ideas in early mathematics, and it's why fact families — the set of addition and subtraction sentences that share the same three numbers — are a natural next step from part-part-whole thinking.
The part-part-whole model also builds a habit of mind that will serve you for a long time: whenever you see a number in a problem, ask "is this a part or a whole?" Knowing which role a number plays tells you which operation to use. If the problem gives you both parts and asks for the whole, you add. If it gives you the whole and one part and asks for the missing piece, you subtract. The model is a thinking tool — a way of organizing what you know and what you need to find, rather than a mechanical procedure to follow.