Sensim Math · Depth 한국어

3-1 · Addition and Subtraction

Sum consecutive numbers via the middle

3.OA.D.93.NBT.A.22.NBT.A.2 · take · grade 3

Archetype: Sum of Evenly Spaced Numbers via the Middle · step in a 5-type progression

▶ Practice — 11 problems

Find the sum of the six consecutive whole numbers below.

251+252+253+254+255+256251 + 252 + 253 + 254 + 255 + 256

Show solution

Understand

We must add six numbers that increase by 1 each time: 251, 252, 253, 254, 255, and 256, and report their total.

Givens
  • The six consecutive whole numbers are 251, 252, 253, 254, 255, 256
  • Each number is exactly 1 more than the one before it
Unknowns
  • The sum of all six numbers
Constraints
  • The numbers are consecutive (step of 1)
  • There are exactly six of them

Plan

#5 Look for a Pattern · also uses: #7 Identify Subproblems

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.

Execute

#5 Look for a Pattern 3.OA.D.9
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=507251+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.
#7 Identify Subproblems 3.OA.D.9
Six numbers split into three pairs, and every pair equals 507. So the total is three copies of 507.
507×3507 \times 3
Turning repeated equal sums into a multiplication is faster and less error-prone than adding one by one.
#7 Identify Subproblems 3.NBT.A.2
Compute 507 times 3 to get the final total.
507×3=1521507 \times 3 = 1521
A small multiplication finishes the job that six separate additions would have done.
Answer: 1521

Review

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.

Use the middle-value idea (tool 9, easier related problem): the average of the six is 253.5, and 253.5 x 6 = 1521, confirming the pairing result.

Standards · min grade 3

  • 3.OA.D.9 Identify arithmetic patterns and explain using properties of operations — Recognizing that outside-in pairs of consecutive numbers are equal
  • 3.NBT.A.2 Fluently add and subtract within 1000 — Adding each pair to 507 and within the total
  • 2.NBT.A.2 Count within 1000, skip-count by 5s, 10s, and 100s — Recognizing the consecutive counting pattern of the six numbers
💡 Pair the ends of consecutive numbers - they all match, so you just multiply one pair by how many pairs there are!