Problem
Sum of consecutive numbers by pairing
Add every number from 1 to 100 the way Gauss did. Pair the smallest with the largest and work inward. There are 100 numbers, an even count, so nothing is left over. Find the total with a multiplication.
Reasoning · Grade 3-2 Finding Sums Using Products
Your answer
Watch out Multiplying 101 by 100 is tempting, but each pair uses two numbers — there are 50 pairs.
How to solve
Strategy Look for a Pattern — Pairing the ends reveals that every pair adds to the same total, turning a long sum into one easy multiplication. I split the work into 'find the pair total' and 'count the pairs'.
1STEP 1
Pair the ends to find one fixed total
1 + 100, 2 + 99 — every pair makes 101.
1 + 100 = 2 + 99 = 3 + 98 = 101
Stepping up on one end and down on the other keeps the pair total unchanged, so all pairs are identical.
3.OA.D.9Look For A PatternPairing the first with the last, the second with the second-last, gives the same total every time.
Why?
As one partner steps up by one the other steps down by one, so every pair lands on the identical total.
🧱Pairing from both ends gives a constant sumIn an evenly spaced list the smallest with the largest, the next with the next, always make the same total, so a long sum becomes a multiplication.
Why?
The step up and the step down are the same size and cancel each other, which is why the pair total never drifts.
🧱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.
2STEP 2
Count how many pairs there are
With 100 numbers there are 50 pairs.
100 ÷ 2 = 50
Splitting 100 numbers into pairs of 2 is just dividing by 2.
3.OA.A.3Identify Subproblems3STEP 3
Multiply pair total by number of pairs
Multiplying gives 50 × 101 = 5050.
50 × 101 = 5050
Fifty equal groups of 101 is multiplication, far quicker than adding one hundred numbers.
3.OA.C.7Look For A PatternAnswer
5050
50 × 101 = 5050
There are 100 numbers averaging about 50 (halfway between 1 and 100, namely 50.5), so the sum should be near 100 × 50.5 = 5050 — exactly what we got. The size (about five thousand) is sensible for adding numbers up to 100.
Takeaway
Pairing the smallest and largest numbers makes every pair the same, so a giant sum becomes one quick multiplication — Grade 3 thinking like Gauss!
- Pair the ends to find one fixed total
- Count how many pairs there are
- Multiply pair total by number of pairs