Problem
Compute one expression several ways
Work out 57 + 28 three ways and 72 − 39 two ways. Every method only reorganizes the same numbers, so the answers must agree. Show all five computations and give the answers.
Reasoning · Grade 2-1 Calculating Cleverly
Your answer
Watch out As in addition it is tempting to nudge only one number to 40. To keep the gap, both numbers must move by 1.
How to solve
Strategy Organize Information in More Ways — The whole point is to re-sort the same numbers in different ways and see that the total never changes. Reorganizing into tens-and-ones (tool 15) is the core move, and a number-line or jump picture (tool 1) shows why shifting a piece from one number to the other keeps the sum the same.
1STEP 1
Addition Method 1 — split the first number
Split 57 into 50 and 7: 50 + 28 = 78, then 78 + 7 = 85.
57 + 28 = (50 + 7) + 28 = (50 + 28) + 7 = 78 + 7 = 85
Splitting a number into its tens and ones and adding the easy part first is exactly the within-100 mental strategy second graders practice.
2.NBT.B.5Organize Information In More Ways2STEP 2
Addition Method 2 — split the second number
Split 28 into 20 and 8: 57 + 20 = 77, then 77 + 8 = 85.
57 + 28 = 57 + (20 + 8) = (57 + 20) + 8 = 77 + 8 = 85
Adding a whole ten (jump of 20) is easy, and crossing from 77 to 85 is a familiar ones-bridge over the next ten.
2.NBT.B.5Organize Information In More Ways3STEP 3
Addition Method 3 — pass the same amount across
Pass 2 from 57 to 28 and the sum holds: 55 + 30 = 85.
57 + 28 = (57 - 3) + (28 + 3) = 54 + 31 = 85
Picture two stacks of cubes: moving cubes from one stack to the other does not change the grand total, so making one number 'round' is free.
2.NBT.B.5Draw A DiagramPassing the same amount from one number across to the other leaves the sum untouched, so you may round one of them for free.
Why?
What one stack loses the other stack gains, so the two changes cancel and the grand total cannot move.
🧱What one part gains the other losesMove some amount from one addend to the other and the total does not budge, so you may make one number round for free.
Why?
The same trick works for subtraction the other way round: slide both numbers up by the same amount and the gap between them is unchanged.
🧱Shift both by the same amount and the difference holdsSlide two points the same distance along the line and the gap between them never changes.
4STEP 4
Subtraction Method 1 — split the number you take away
Split 39 into 30 and 9: 72 − 30 = 42, then 42 − 9 = 33.
72 - 39 = 72 - (30 + 9) = (72 - 30) - 9 = 42 - 9 = 33
Subtracting a full ten is easy, and 42 - 9 is a single ten-bridge a second grader can do mentally.
2.NBT.B.5Organize Information In More Ways5STEP 5
Subtraction Method 2 — keep the gap the same
Nudge both up by 1 and the gap holds: 73 − 40 = 33.
72 - 39 = (72 + 1) - (39 + 1) = 73 - 40 = 33
On a number line the distance between two points does not change when you slide both points the same amount, so rounding 39 up to 40 keeps the difference 33.
2.NBT.B.5Draw A DiagramAnswer
57 + 28 = 85, 72 − 39 = 33
All three addition methods give 85 and both subtraction methods give 33, so the answers agree. The standard check confirms: 57 + 28 = 85 (7+8=15, carry 1; 5+2+1=8) and 72 - 39 = 33 (72 is 33 more than 39 since 39+33=72). Magnitudes are sensible: 85 is a bit under 90, and 33 is about a third of 72.
Takeaway
There are many friendly paths to the same total — this only needs the Grade 2 tens-and-ones thinking you already use!
- Addition Method 1 — split the first number
- Addition Method 2 — split the second number
- Addition Method 3 — add and subtract the same amount
- Subtraction Method 1 — split the number you take away
- Subtraction Method 2 — keep the gap the same