Problem
Discover the hidden operation rule
A machine changes how many items come out by always applying one hidden rule. Put in 3 and 7 come out; 2 gives 5; 4 gives 9; 7 gives 15; 6 gives 13. Find the rule and say how many come out when 8 go in.
Reasoning · Grade 2-2 Rules and Problem Solving
Your answer
Watch out One row like 3 → 7 suggests 'add 4', but a rule has to fit all five rows.
How to solve
Strategy Look for a Pattern — Each row pairs an input with an output, so we look for one rule that turns every input into its matching output. Comparing the pairs shows the output is a bit more than double the input; we guess 'double and add 1' and check it against all five rows before using it on 8.
1STEP 1
Compare each output to its input
The gaps are not constant, so adding alone cannot be the rule.
7-3=4, 5-2=3, 9-4=5, 15-7=8, 13-6=7
Subtracting input from output is how a Grade 2 student first checks 'how much more' and notices the gap keeps growing as the input grows.
2.NBT.B.5Look For A Pattern2STEP 2
Notice the output is about double the input
Each output is one past double the input: double, then add 1.
3 × 2=6, 2 × 2=4, 4 × 2=8
Doubling is just adding a number to itself, which Grade 2 students do with equal groups of two.
2.OA.C.4Look For A Pattern3STEP 3
Check the rule on every row
Tested against all five rows, every one fits.
2 × 7+1=15, 2 × 6+1=13
Trying the guessed rule on each known pair is the natural way to be sure before trusting it on a new number.
2.OA.B.2Guess And CheckA guessed rule is trustworthy only once it has been tried on every row of the table and has fitted them all.
Why?
A single row where the rule gives the wrong output is enough to prove the rule wrong, so every row has to be tested.
🧱One counterexample refutes an always-claimA claim about every case dies the moment you exhibit one case where it fails, and exhibiting is easier than arguing.
Why?
The rule doubles the input, which is taking two equal groups of it, so the same doubling has to work whatever the input is.
🧱Multiplication as equal groupsa times b means a equal groups of b — it is just repeated adding of the same amount.
4STEP 4
Apply the rule to 8 erasers
Putting in 8 gives 2 × 8 + 1 = 17.
2 × 8+1=16+1=17
Once the rule is known, finding the new output is one doubling and one addition a Grade 2 student can do mentally.
2.OA.B.2Guess And CheckAnswer
17 erasers
2 × 8 + 1 = 17
8 is between the inputs 7 (which gives 15) and would-be 9, so an output of 17 sits just above 15, exactly as 'double and add 1' predicts. The answer is an odd number, which fits because doubling always gives an even number and adding 1 makes it odd, matching every odd output in the table.
Takeaway
This only needs the Grade 2 idea of doubling and adding 1 -- find the rule once and every answer pops right out!
- Compare each output to its input
- Notice the output is about double the input
- Check the rule on every row
- Apply the rule to 8 erasers