Reasoning · Grade 3-2 Completing Equations
Problem
The 1 in the tens shows the first step leaves remainder 1.
In long division the leftover from one step becomes the front digit of the next number you work with, so that tens digit 1 tells me the first remainder is 1.
4.NBT.B.6Make A Systematic ListIn the 30s, one past a multiple of 6 gives 31 and 37.
Only the numbers in the thirties that sit exactly one above a multiple of 6 can leave the needed remainder 1.
4.NBT.B.6Make A Systematic ListA final remainder of 4 means the brought-down number is 4 past a multiple of 6.
The last line of the long division is the leftover, so 1□ divided by 6 must leave exactly 4.
4.NBT.B.6Make A Systematic ListIn the teens those are 10 and 16, so the ones digit is 0 or 6.
Only the numbers in the teens that sit exactly four above a multiple of 6 can leave the needed remainder 4.
4.NBT.B.6Make A Systematic ListCombining gives 310, 316, 370, 376.
The tens digit and the ones digit are decided by separate steps, so I pair every allowed tens choice with every allowed ones choice.
4.NBT.B.6Guess And CheckThe tens digit and the ones digit are settled by separate division steps, so every allowed tens choice pairs with every allowed ones choice.
Long division works one place at a time and passes only the leftover downward, so the two steps constrain different digits.
Because neither choice limits the other, all the pairings really occur and the counts simply multiply.
Each step of long division leaves a leftover that becomes the next digit you work with — read those leftovers and the hidden digits pop right out!