Problem
Find the rule linking two numbers
One person calls a number and the other answers by a fixed secret rule. The rule never changes. The table records five rounds. Find the answer when 7 is called.
Reasoning · Grade 5-1 Finding the Hidden Rule
Your answer
Watch out One operation seems enough, but only multiply then subtract fits all five.
How to solve
Strategy Look for a Pattern — The table is deliberately shuffled, so the first job is to re-sort the pairs by Mia's number. Once they sit in order the pattern in Noah's answers can be seen. Then I test the simple one-step rules first, add the same number, or multiply by the same number, and show that neither one fits every pair. Ruling those out is what forces a two-step rule. To find the two steps I look at how much Noah's answer jumps when Mia's number goes up by 1, which reveals the multiplying part, and then compare with Noah's actual answers to find the amount that must be taken away. Finally I check the finished rule against all five recorded pairs before I trust it on the sixth.
1STEP 1
Re-sort the table by Mia's number
Re-sort the table by the number called.
(1, 1), (2, 4), (5, 13), (6, 16), (8, 22)
Putting two matching lists side by side in order is the ordinary way to spot how one column depends on the other, and it costs nothing but a rewrite.
5.OA.B.3Organize Information In More Ways2STEP 2
Show that one single operation cannot work
A single operation does not fit.
1 + 0 = 1 but 2 + 0 = 2 ≠ 4; 1 × 1 = 1 but 2 × 1 = 2 ≠ 4
Testing the two easiest rules first takes only a moment, and failing them is useful information: it tells the solver to stop hunting for a one-step answer.
4.OA.C.5Guess And Check3STEP 3
Find the multiplying part from the steps of 1
Each step of 1 raises the answer by 3.
4 - 1 = 3, 16 - 13 = 3
The size of the jump for a step of 1 is exactly the number you are multiplying by, so one clean pair of neighbouring rows hands over the multiplier.
5.OA.B.3Look For A PatternThe size of the jump when the input rises by one is exactly the number being multiplied by.
Why?
Because every step of one produces the same jump, the outputs climb by a fixed amount and that amount is the multiplier.
🧱A constant step reaches any termWhen a list climbs by the same amount every time, any term is the starting value plus that step repeated, so you never have to write out the ones between.
Why?
A step of two must then produce twice the jump, so the multiplying part scales with the input just as it should.
🧱Proportional quantities scale togetherWhen two quantities keep a fixed ratio, doubling one doubles the other, so a ratio you know can stand in for one you do not.
4STEP 4
Find the amount that is taken away
The product always has 2 taken off.
3 - 1 = 2, 6 - 4 = 2, 15 - 13 = 2, 18 - 16 = 2, 24 - 22 = 2
Once the multiplying part is known, comparing what it predicts with what actually happened leaves a single leftover number, and that leftover is the second step of the rule.
4.OA.C.5Look For A Pattern5STEP 5
Check the rule on every recorded pair
The rule fits all five pairs.
1 × 3 - 2 = 1, 5 × 3 - 2 = 13, 2 × 3 - 2 = 4, 6 × 3 - 2 = 16, 8 × 3 - 2 = 22
A rule that matches every row is far safer than one that matches the first row you tried, and checking all five is only five short multiplications.
3.OA.C.7Make A Systematic List6STEP 6
Apply the rule to Mia's 7
Applied to 7 it gives 19.
7 × 3 = 21, 21 - 2 = 19
Once a rule has survived every case you can test, using it on a new number is just following the same two steps in the same order.
4.OA.C.5Look For A PatternAnswer
19
7 × 3 − 2 = 19
The answer sits exactly where it should. Mia's 7 lies between her 6 and her 8, so Noah's answer must lie between 16 and 22, and 19 does. It is also 3 more than 16 and 3 less than 22, matching the steady climb of 3 for every 1 that Mia adds. The answer is a whole number, like every other entry in the table, and it is larger than 7, which fits the pattern that Noah's numbers grow away from Mia's as her numbers get bigger.
Takeaway
When one operation will not fit the table, look at how much the answer jumps for each step of 1: that jump is what you multiply by, and the leftover is what you add or take away.
- Re-sort the table by Mia's number
- Show that one single operation cannot work
- Find the multiplying part from the steps of 1
- Find the amount that is taken away
- Check the rule on every recorded pair
- Apply the rule to Mia's 7