A number goes in, each box does something to it, and a new number comes out. Run the machine, run it backwards, then crack the secret rule.
Put the number through the box. What comes out?
A number goes in, each box changes it, and a new number comes out.
Two boxes? Do them in order, left to right.
Going backwards: undo the last box first, with the opposite.
To find the rule, see how much out grows when in grows by 1.
+ and − undo each other.
× and ÷ undo each other.
A picture of a rule: a number goes in, each box does something to it, like × 3 or + 1, and a new number comes out. It helps children see a calculation as a set of steps, and it leads straight into algebra.
Start with the number that came out and undo each box from the last to the first, using the opposite operation. If the machine does × 3 then + 1 and 16 comes out: 16 − 1 = 15, then 15 ÷ 3 = 5.
Put in numbers that go up by 1 and see how much the output goes up: that is the number to multiply by. Then see what has to be added or taken away to fit. Always check the rule on every pair.
Children use one-step machines at about 7 to 9, two-step machines and working backwards at about 9 to 11, and finding the rule leads into algebra at about 10 to 12.