Problem
10+11+12=33 is 3×11, and 8+9+10+11+12=50 is 5×10 — for an odd run, the sum equals the count times the middle number.
Pairing the ends inward, each pair averages the middle, so the whole sum centers on the middle.
3.OA.D.9Look For A PatternFor an odd run of consecutive whole numbers, the sum equals how many numbers there are times the middle number.
You can move the surplus from each number above the middle down to its matching number below, turning every number into the middle without changing the total.
A number a few steps above the middle is over it by the same amount that its matching number the same steps below is under it, because stepping up by some amount and then stepping back down by that same amount cancel out.
Sliding those equal amounts from the high numbers into the low ones only rearranges the same units, so the total of all the numbers stays the same.
Once every number has become the middle, the sum is that many equal middles added together, and adding one equal amount over and over is the same as the count times the middle.
Five consecutive numbers sum to 35, so the middle is 35 divided by 5 = 7. The run is the two before and two after 7.
Knowing the middle, just step out two on each side.
4.OA.C.5Look For A PatternSeven consecutive numbers sum to 140, so the middle is 140 divided by 7 = 20. The run is the three before and three after 20.
The middle is the balance point, so spread three numbers to each side.
4.OA.C.5Look For A PatternThis only needs Grade 4 pattern sense: divide the sum by how many numbers to find the middle, then count outward!