Problem
Sum consecutive numbers via the middle
We must add six numbers that increase by 1 each time: 251, 252, 253, 254, 255, and 256, and report their total.
OperationsBase-ten numbers
Your answer
Watch out Don't stop at one pair's sum of 507 — that's just one of three equal pairs, not the total.
How to solve
Strategy Look for a Pattern — Consecutive numbers have a balancing pattern: pairing the smallest with the largest gives equal partial sums. Spotting this turns six additions into a simple multiplication.
1STEP 1
Pair from the outside in
Pair the first with the last, the second with the fifth, and the third with the fourth. Each pair adds to the same total.
251+256 = 507, 252+255 = 507, 253+254 = 507
When numbers go up by 1, what you add to one end you take from the other, so the outside-in pairs stay equal.
3.OA.D.9Look For A PatternPairing the six numbers from the outside in — first with last, second with fifth, third with fourth — makes pairs that each add up to the same total of 507.
Why?
Stepping from one pair to the next pair inside it, the smaller number goes up by 1 while the larger number goes down by 1, since the six numbers increase by exactly 1 each time.
Why?
A rise of 1 on one number and a drop of 1 on the other are opposite changes, so the pair's total does not move.
Why?
Adding 1 and then taking that 1 back away lands you exactly where you started, because taking away reverses adding.
🧱Inverse operationsSubtraction undoes addition and division undoes multiplication — they reverse each other.
Why?
The gain and the equal loss come to a change of zero, and adding zero to the total leaves it exactly the same.
🧱Additive identity (zero)Adding zero adds nothing, so the number stays exactly the same.
2STEP 2
Count the equal pairs
Six numbers split into three pairs, and every pair equals 507. So the total is three copies of 507.
507 × 3
Turning repeated equal sums into a multiplication is faster and less error-prone than adding one by one.
3.OA.D.9Identify Subproblems3STEP 3
Multiply to get the sum
Compute 507 times 3 to get the final total.
507 × 3 = 1521
A small multiplication finishes the job that six separate additions would have done.
3.NBT.A.2Identify SubproblemsAnswer
1521
507 × 3 = 1521
The six numbers cluster around 253.5, so the sum should be near 6 x 253.5 = 1521. Our answer 1521 matches exactly, and it is an even total which fits adding six numbers around 253. The magnitude is sensible.
Takeaway
Pair the ends of consecutive numbers - they all match, so you just multiply one pair by how many pairs there are!
- Pair from the outside in
- Count the equal pairs
- Multiply to get the sum
Where next?
Level upsuggested
Another one like thissuggested
Step back and solidifysuggested