Here is a number machine with a rule inside it: × 3. When you put 4 in at the top, what number comes out the bottom? And what about 7?

Watch four different machines, each with its own rule. The rule sits inside the machine and tells it what to do to every number you put in. The table beside each machine records the input-output pairs as they happen.

Example 1: the × 3 rule

Watch the table fill in: 4 goes in and 12 comes out, 7 goes in and 21 comes out, 10 goes in and 30 comes out, 0 goes in and 0 comes out.

Example 2: the + 5 rule

The rule changes. Now every input has 5 added on. Watch how the gap between input and output stays the same on every row.

Example 3: a two-step rule, × 2 then + 1

This machine does two things in a row. First it doubles the input; then it adds 1. So when 3 goes in, the machine doubles it to 6, then adds 1, and 7 comes out.

Now watch what would happen if the machine did the two steps in the opposite order. If it added 1 first, 3 becomes 4. Then it doubles, and 8 comes out instead of 7. Same numbers, different order, different answer. That's why the order matters.

Example 4: × 10 with a decimal input

The same machine works on decimals too. When you put 0.5 into a × 10 machine, you get 5 out.

Now you drive the × 3 machine together. You choose an input, the machine sends it through the × 3 rule, and the input-output pair lands in the table. The class watches the table grow, row by row, into the table of values for the × 3 rule.