Patterns and rules · Usually met around P4
Number machines
Work backwards, doing the opposite at each step.
How to do it
Method
Work backwards, doing the opposite at each step
× 3 + 1 gives 22: 22 − 1 = 21, 21 ÷ 3 = 7
Another way
Put it back in
7 × 3 + 1 = 22 ✓
In full
A number machine turns every number put in into a number out by the same rule, say “× 3, then + 1”. To find the output, work in order: × 3 first, then + 1. To find the input, work backwards from the last step, doing the opposite each time: − 1 first, then ÷ 3. To find the rule, see how much the output goes up when the input goes up by 1: that is what to multiply by. Then see what is still missing to add or take away. Check the rule on every pair in the table: it is right only if every pair fits.
Worked examples
Have a go first, then open each one for the answer and the working.
Example 1
Each input in the table goes to its output by the same rule. What is the ? in the table?
Show the answer and working
Answer: 30
Each time the input goes up 1, the output goes up 3: so × 3. 1 × 3 is 3 short: so + 3. 9 × 3 + 3 = 30.
Example 2
Every input and output in the table follow the same rule. Which rule is it?
- A. × 5, then + 4
- B. × 9
- C. + 8
- D. × 6, then + 3
Choose the right answer
Show the answer and working
Answer: A. × 5, then + 4
The output goes up 5 each time: × 5. 1 × 5 = 5, 4 short: + 4. Every pair fits.
Example 3
Each input in the table goes to its output by the same rule. Which input gives the output 26?
Show the answer and working
Answer: 8
The rule is “× 3, then + 2”. Work backwards: 26 − 2 = 24, 24 ÷ 3 = 8.
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