Problem
Fast sums using pattern and symmetry
Find the sum 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8. Instead of adding left to right, pair the numbers from the two ends inward. Every pair turns out to have the same sum. Get the total with one multiplication.
Reasoning · Grade 2-2 Applying the Multiplication Facts
Your answer
Watch out Eight numbers suggest eight pairs, but pairing takes two at a time — there are 4.
How to solve
Strategy Look for a Pattern — I Look for a Pattern in the pair sums: pairing from the ends inward, each pair adds to the same total. Drawing the numbers in a row with arcs joining the outer pairs makes the equal sums visible. Once I see all pairs share one sum, the total is just that sum times the number of pairs - a single multiplication.
1STEP 1
Pair the numbers from the ends inward
Pairing from the ends inward makes 4 pairs.
(1,8), (2,7), (3,6), (4,5)
Folding the list in half lines up each small number with a large one, giving neat pairs.
2.OA.B.2Draw A Diagram2STEP 2
See that every pair sums to the same number
1+8, 2+7, 3+6, 4+5 all give 9.
1+8 = 2+7 = 3+6 = 4+5 = 9
One side rising while the other falls by the same amount keeps the total fixed - that is the pattern that makes every pair equal.
2.OA.B.2Look For A PatternEvery pair taken from the two ends inward adds to the very same total.
Why?
As you move inward, one partner rises by a step while the other falls by the same step, so the pair total is fixed all the way.
🧱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?
What one partner gains the other gives up, and those two equal changes cancel each other exactly.
🧱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.
3STEP 3
Multiply the equal pair sums
So the total is 4 × 9 = 36.
4 × 9 = 36
Equal pair sums turn a long addition into 'how many pairs times the pair sum,' which is one quick multiplication.
3.OA.A.1Look For A PatternAnswer
36
4 × 9 = 36
Adding straight across as a check: 1+2+3+4+5+6+7+8 = (1+2+3+4) + (5+6+7+8) = 10 + 26 = 36, which matches 4 x 9 = 36. The answer 36 is also sensible in size: 8 numbers averaging about 4.5 should total around 8 x 4.5 = 36.
Takeaway
Pair the ends inward, notice each pair makes 9, then multiply: 4 pairs x 9 = 36!
- Pair the numbers from the ends inward
- See that every pair sums to the same number
- Multiply the equal pair sums