Number Bonds: The Part-Part-Whole Model

What a number bond shows

The diagram has three circles. The whole sits at the top (or the center), and two parts branch off it. Any two of the three numbers determine the third, and that is the source of the tool's power: a single picture holds an entire family of facts. The bond for 8, 5, and 3 contains 5 + 3 = 8, 3 + 5 = 8, 8 - 3 = 5, and 8 - 5 = 3 — four equations in one shape. Children learn to read the picture before they learn to write any of the four, which is exactly the right order.

The three questions a bond can ask

Cover any one of the three circles and you get a different kind of practice. There are exactly three:

• Hide a part: you see the whole and one part, and find the missing part. This is subtraction, met as a find-the-part question rather than a take-away.
• Hide the whole: you see both parts, and find the whole. This is addition in its plainest form.
• Hide nothing, choose the parts: you are given only the whole and asked for any pair that makes it. This is decomposition — and it is the one that builds flexibility, because there is more than one right answer.

That third question is the one most worth practising. A child who can only answer 8 = 5 + 3 has memorized a fact; a child who can rattle off 8 as 4 and 4, 6 and 2, 7 and 1, and 8 and 0 has understood that a number is a quantity that can be taken apart and put back together however the problem needs. That flexibility is what mental math is made of. You can practise all three directly on number bonds worksheets.

Bonds of 10 — the milestone

Of all the bonds a child learns, the pairs that make ten earn the most attention. Knowing instantly that 7 needs 3 more to reach ten, or that 6 and 4 are partners, is one of the highest-leverage facts in early math — it powers the making-ten strategy, bridging through ten in addition, and mental subtraction later on. It is worth practising to the point of automaticity, the way a musician practises scales.

Six pairs (read each one both ways and you have all eleven combinations). A child who knows these cold can add 8 + 5 by taking 2 from the 5 to make 8 into 10, then adding the leftover 3 — the making-ten strategy in one move. None of that is available until the bonds of ten are second nature.

How to teach them: concrete, pictorial, abstract

Number bonds are usually introduced in kindergarten and first grade, and they teach best in the standard three-stage progression that moves from things to pictures to symbols:

1. Concrete. Give the child a handful of counters — buttons, cubes, pasta — and have them split a group of eight into two piles, again and again, saying what they made each time. The splitting is the concept, done with their hands.
2. Pictorial. Move to the bond diagram: the same split, now drawn as a whole and two parts. The picture is the bridge between the physical piles and the written equation.
3. Abstract. Finally, write the number sentences the bond contains. By now the equation is a caption for something the child already understands, not a rule handed down from nowhere.

Rushing to the abstract stage is the common mistake, and it produces children who can copy an equation but cannot explain it. Let the counters and the diagrams do their work first. Once the relationship is solid, mixed practice on addition worksheets and subtraction worksheets turns the derived facts into remembered ones.

Number bonds and fact families

A number bond and a fact family are two views of the same thing. The bond is the picture — the whole and its two parts; the fact family is the four equations that picture generates. Teaching the bond first means the fact family is not a list to memorize but a set of sentences the child can read straight off the diagram. The same relationship reappears later in place value (a two-digit number is a bond of tens and ones) and stands behind the patterns in skip counting, which is why the bond is worth getting right early.

Frequently asked questions