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?
Sketch a quick box-with-an-arrow-in-and-arrow-out on the IWB with × 3 in the middle. Take three hands-up answers for input 4, then a second round of hands-up for input 7. Five seconds of quiet think-time before any hands go up.
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.
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.
The rule changes. Now every input has 5 added on. Watch how the gap between input and output stays the same on every row.
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.
The same machine works on decimals too. When you put 0.5 into a × 10 machine, you get 5 out.
Walk each example aloud, one at a time. On Example 1, point to the 0 → 0 row and say even zero obeys the rule. On Example 2, point along a column and ask 'what's the same on every row?'; the gap of 5 is the rule made visible.
Between Example 2 and Example 3, pause for five seconds and ask the class to name what the + 5 machine did, in one short sentence each. Take two hands-up answers and revoice the better one before opening Example 3. That brief pause keeps the watching class from filling up before the two-step rule lands.
On Example 3, narrate the two steps in order: double it first, then add 1. The interactive shows the × 2 then + 1 path; the reversed-order contrast (+ 1 then × 2, giving 8 instead of 7) sits in the description for pupils to read, so anchor it by saying it aloud: same numbers, different order, different answer. On Example 4, pause on 0.05 → 0.5; this is the place-value-shift moment from earlier in the year. A × 10 machine works on decimals too.
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.
This round is for talking it through together; pupils take turns at the board and the class agrees or corrects out loud.
Bring four pupils up in turn, and give each turn a distinct job so the watching class always has something new to notice:
In your maths copy, draw two input/output tables. Write the rule at the top of each table, then fill in four input-output pairs underneath, using the inputs given.
Walk the room glancing at the rule label at the top of each table and the alignment of the in/out columns. No marking — this is whole-class copybook practice, not assessment. The inputs are specified on the IWB so pupils don't have to invent them; everyone is working on the same eight calculations and the pace stays even.
If a pupil finishes early, they can add one more row to either table with an input of their own choosing.
Today's challenge: build the table of values for five different rules in turn. The rule is shown at the top of each machine; your job is to work out what comes out for each of the four given inputs.
One pupil at the board per challenge, in turn. Read the rule aloud as a class before the first input is typed: 'this machine multiplies by 3'. After each output, the class confirms or corrects out loud, then the next pupil comes up for the next challenge.
Look for pupils slipping on the two-step rule; listen for someone saying 'add 1 first then double' and revoice double first, THEN add 1. The × 10 challenge with decimals (0.5, 1.2, 4, 0.05) is the trickiest of the core bank; anchor it back to the place-value-shift work earlier in the year if anyone wobbles.
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