Why a number bond is worth four sums
Children who learn 4 + 3 and 7 − 3 as separate facts are carrying twice the load, for the rest of their schooling.
Ask a child what seven take away three is, straight after they have told you four and three make seven, and a good number of them will start counting again. They are not being careless. Nobody has shown them that it is the same fact.
A number bond holds two parts and a whole together. Seven is four and three — and that one picture answers 4 + 3, 3 + 4, 7 − 3 and 7 − 4. One thing to know, not four to remember.
The same picture does multiplication
Put a product on top and a factor pair underneath and nothing else changes: 12 is 3 and 4, so 3 × 4, 4 × 3, 12 ÷ 3 and 12 ÷ 4 are again one fact seen from four sides. Division is not a new picture. It is this one, read downwards.
Bond Rush drills exactly this — sixty seconds, three lives, or no clock at all, on either family. Get far enough and a part splits again, which is where making ten appears: 8 + 7 is 8 + 2 + 5, because the seven was split to fill the eight up to ten first.
This is the picture the whole Singapore method is built on. The bond becomes a bar, the bar becomes a comparison, and the comparison becomes a unit standing for something nobody has told you — which is algebra, arriving without being announced.