What's my rule?
Tonight's idea
Think of a machine. A number goes in, the machine does something to it, and a number comes out. The machine follows a rule, and it uses the same rule every time.
If nobody tells you the rule, you can find it from a table of ins and outs. Watch what happens to the out when the in goes up by 1.
If the out goes up by 2 each time, the rule has × 2 in it. Up by 5 each time means × 5.
Then test one number to see what is added or taken away at the end.
A machine can also run backwards. Start with the out and undo each step, last step first. Adding is undone by subtracting. Multiplying is undone by dividing. It is like socks and shoes: shoes go on last, so they come off first.
Watch one: find the rule, then go backwards
| in | 1 | 2 | 3 | 4 |
| out | 5 | 7 | 9 | 11 |
The out goes up by 2 each time, so the rule has × 2. Test the first column: \(1 \times 2 = 2\), but the out is 5. That is 3 more. The rule is × 2, then + 3. Check the last column: \(4 \times 2 + 3 = 11\).
Now backwards. The out is 21. What went in? Undo the + 3: \(21 - 3 = 18\). Undo the × 2: \(18 \div 2 = 9\). The in was 9.
Your turn
- A machine's rule is "× 4, then − 1". What comes out when you put in (a) 2, (b) 5, (c) 10?
- Find the rule for this table. Then say what comes out when 10 goes in.
in 1 2 3 4 out 4 7 10 13 - A machine's rule is "× 5, then + 2". (a) 47 comes out. What went in? (b) 102 comes out. What went in? (c) Only whole numbers go in. Can 30 ever come out? Say why.
- This table has gaps. The ins do not go up by 1.
(a) From 2 to 5 the in goes up by 3. How much does the out go up? So how much is that for each 1? (b) Find the rule. (c) What goes in to make 101 come out?in 2 5 10 out 9 21 41
Stretch: two machines, one job
Machine A is "× 2, then + 6". Machine B is "+ 3, then × 2". Try three numbers in each. What do you notice? Explain why it must happen for every number.
Answers
- (a) \(2 \times 4 - 1 = 7\). (b) \(5 \times 4 - 1 = 19\). (c) \(10 \times 4 - 1 = 39\).
- The out goes up by 3 each time, so × 3. Then \(1 \times 3 = 3\) and the out is 4, so + 1. The rule is "× 3, then + 1". For 10: \(10 \times 3 + 1 = 31\).
- (a) \(47 - 2 = 45\), then \(45 \div 5 = 9\). (b) \(102 - 2 = 100\), then \(100 \div 5 = 20\). (c) No. Going backwards, \(30 - 2 = 28\), and 28 does not divide exactly by 5. Every out is 2 more than a number in the 5 times table.
- (a) The out goes up by \(21 - 9 = 12\). That is \(12 \div 3 = 4\) for each 1. (b) So × 4. Test: \(2 \times 4 = 8\) and the out is 9, so + 1. The rule is "× 4, then + 1". Check: \(10 \times 4 + 1 = 41\). (c) \(101 - 1 = 100\), then \(100 \div 4 = 25\).
Stretch. They always agree. For example 1 gives 8, 4 gives 14 and 10 gives 26 in both. Machine B doubles the number and also doubles the 3 it added. Double 3 is 6, so B gives "double the number, plus 6". That is machine A. With brackets: (number + 3) × 2 = number × 2 + 6.