Reasoning · Grade 3-1 Setting Up the Equation
Problem
Write it as hundreds, tens and ones; the reversal just swaps the first and last digits.
Naming each digit by its place is just saying a three-digit number is so many hundreds, tens, and ones - the heart of place value.
2.NBT.A.1Convert To AlgebraSubtracting cancels the tens and leaves 99 times the gap, so first minus last is 4.
When you reverse, only the hundreds and ones trade places, so the gap depends only on how far apart those two digits are.
3.NBT.A.2Convert To AlgebraReversing a three-digit number leaves the tens digit alone, so the difference depends only on how far apart the outer two digits are.
Swapping the hundreds digit with the ones digit trades a count of hundreds for a count of ones and back, while the tens count never moves.
Each number is its three place amounts added together, so a part that is identical in both simply cancels when you subtract.
With the last digit at least 1, there are 5 pairs.
Reversing must give a real three-digit number, so the ones digit can't be 0; that throws out the (4,0) pair and leaves 5.
2.NBT.B.5Make A Systematic ListThe middle digit has 8 choices each: 5 × 8 = 40.
Five choices for the outer digits and eight independent choices for the middle digit multiply, just like rows times columns.
3.OA.A.1Make A Systematic ListReversing only swaps the outer digits, so the whole puzzle shrinks to 'how far apart are the first and last digits?' - Grade 3 place value handles it!