The Number Machine and the Rule That Fit Once
Mrs. Ibarra's math center has a number machine. A card goes in, the machine follows one hidden rule, and a card comes out. Three results are taped to the board:
in 4 → out 11 in 6 → out 15 in 9 → out ?
Tomas says the rule is "add 7," because 4 + 7 = 11. Ravi says the rule is "double the number, then add 3," because 4 × 2 + 3 = 11 and 6 × 2 + 3 = 15. Mrs. Ibarra writes a second challenge on the board - which card must go in to get 27 out? - and a third: "Maya says 'double then add 3' and 'add the number to itself, then add 3' are two different rules. Who is right?" Students may use counters and a hundred chart, but not a calculator.
1. Which rule is correct, and what comes out when 9 goes in? Show the test that settles Tomas against Ravi.
Read the worked answer
Ravi's rule is correct: double, then add 3. The output for 9 is 21.
The test is to run each proposed rule on every row, not just the first one.
Tomas - add 7:
- 4 + 7 = 11 ✔
- 6 + 7 = 13 ✘ (the machine showed 15)
One match is not a rule. Tomas's rule is dead after the second card.
Ravi - double, then add 3:
- 4 × 2 + 3 = 11 ✔
- 6 × 2 + 3 = 15 ✔
- So 9 × 2 + 3 = 21 ✔
A rule has to explain the cards you already have before it earns the right to predict the next one. The fact that Tomas's rule worked on 4 is exactly why the machine shows more than one card.
2. Find the card that goes in to get 27 out. Show each undo step, and explain why the steps cannot be done in the other order.
Read the worked answer
The input is 12. Work backwards, undoing one step at a time.
Forward rule: input × 2, then + 3 → output. Backwards: output − 3, then ÷ 2.
- 27 − 3 = 24
- 24 ÷ 2 = 12
- Check: 12 × 2 = 24, 24 + 3 = 27 ✔
Why the order cannot flip: in the forward direction the doubling happens before the adding, so the +3 never gets doubled. If you undo in the wrong order you get 27 ÷ 2 = 13.5, then 13.5 − 3 = 10.5, which is not a whole card and does not check.
A quick way to remember it: undo in the reverse of the order things were done, the way you take off shoes before socks.
3. Maya says "double then add 3" and "add the number to itself then add 3" are two different rules. Is she right? Give evidence a classmate could check.
Read the worked answer
Maya is not right - they are the same rule said two ways.
Doubling a number means adding it to itself. The word "double" is a shortcut for n + n.
Check with test cards:
| in | n + n + 3 | n × 2 + 3 |
|---|---|---|
| 4 | 4 + 4 + 3 = 11 | 8 + 3 = 11 |
| 6 | 6 + 6 + 3 = 15 | 12 + 3 = 15 |
| 9 | 9 + 9 + 3 = 21 | 18 + 3 = 21 |
Same output on every card, so it is one rule. This is a different situation from Tomas's "add 7." Tomas's rule gave the right answer on one card and then disagreed. Two descriptions that agree on every input are the same rule, even when the words sound different.